Why can a real number be defined as a Dedekind cut, that is, as a set of rational numbers? I don't know if my textbook is written poorly or I'm dumb. But I can't bring myself to understand the following definition.

A real number is a cut, which parts the rational numbers into two classes. Let $\mathbb{R}$ be the set of cuts. A cut is a set of rational numbers $A \subset \mathbb{Q}$ with the following properties:
i) $A \neq \emptyset$ and $A \neq \mathbb{Q}$.
ii) if $p \in A$ and $q < p$ then $q \in A$.
iii) if $p \in A$, there exists some $r \in A$ so that $p < r$ (i.e. $A$ doesn't contain the "biggest" number).

That's a literal translation from my textbook (which is written in Slovenian). All seems fine and I can get my head around all of the postulations except for one. The definition states in the beginning "A real number is a cut...", but then it also states "A cut is a set of rational numbers..." So a real number is 'a set of rational numbers'?!
It's not my bad translation, I swear, I'm quite good at English. Either the textbook is written in such a convoluted manner that I can't properly understand the wording the author chose or I'm overlooking something big. Could you please clarify and explain the definition in full detail?
 A: I think you should begin by asking a simpler question : what is a rational number? Or perhaps even simpler: what is an integer or a natural number?
Let's take the existence of natural numbers for granted (doing this helps us avoid lot of set theory). Then an integer can be described by a single natural number and the idea of a negative. Going forward one can describe a rational number using two integers. Note that it is not possible to describe a rational number using just one integer and the idea of a reciprocal.
The next step of describing a real number in terms of rationals is difficult and most presentations focus on the set theoretic aspect rather than the essence of the matter. The key idea is that a real number can not be described in terms of a finite number of rationals. Rather to describe a real number we require an infinite number of rationals in an essential manner. The method of describing the real number may be different (like infinite decimal representation, Dedekind cut, Cauchy sequence etc) but all the descriptions have one thing in common. They all deal with an infinite number of rationals. And this is the primary source of difficulty in understanding the nature of a real number. The most natural way to handle the situation is to define a real number as some infinite set of rationals with specific properties.
Without going into a specific definition of a real number, it should be noted that the rationals being used to describe a real number are approximations to the real number being defined and it is essential that the description includes as good approximation as needed. To put the matter in crude terms, real numbers are things that are well approximated using rationals and we usually identify the real number with the set of rationals approximating it. 
A: As I said in my comment, you are in good company---in fact, the company of Dedekind himself! In a letter to Heinrich Weber, Dedekind says the following:

(...) I would advise that by [natural] number one understand not the class itself (...) but something new (corresponding to this class) which the mind creates. (...) This is precisely the same question that you raise at the end of your letter in connection with my theory of irrationals, where you say that the irrational number is nothing other than the cut itself, while I prefer to create something new (different from the cut) that corresponds to the cut and of which I prefer to say it brings forth, creates the cut. (Ewald, From Kant to Hilbert, vol. 2, p. 835)

So Dedekind himself preferred not to identify the real number with the cut, merely saying that the mind somehow creates the real number which then corresponds to the cut. This is, however, a little bit obscure, so it's not surprising that most mathematicians (such as Weber!) decided to ignore Dedekind's suggestion and simply identify the real number with the cut. The reasoning behind this identification is roughly the following.
We know that any Dedekind-complete ordered field is isomorphic to the field of the real numbers. In particular, this means that any construction or theorem carried out in the real numbers could be reproduced inside an arbitrary Dedekind-ordered field, and vice-versa, by simply using the isomorphism as a "translation" between the fields. Hence, it doesn't matter what the real numbers actually are; for mathematical purposes, even supposing that there is such a thing as the real numbers, anything that we wanted to do with them could also be accomplished in an arbitrary Dedekind-complete ordered field.
Thus, if we could show that the cuts themselves satisfy the axioms for being a Dedekind-complete ordered field, then we could dispense with the real numbers altogether and simply work with the cuts themselves. And, in fact, we can show that this is the case! One need only to show that, given two cuts, $X$ and $Y$, it's possible to define operations on them corresponding to the usual operations on the real numbers, such as addition and multiplication, and that after doing so these operations will satisfy the field axioms. It's not difficult to see that the obvious operations will yield the desired result (exercise!), though it is somewhat laborious. If you are interested in seeing a detailed verification, I recommend reading, say, Appendix A of Yiannis Moshovakis excellent book Notes on Set Theory, which contains a very thorough discussion of the matter.
A: Based on the comments, I think you have a good intuitive grasp of the intent of the definition already.
Namely, when we choose to use Dedekind cuts to define the reals,
the idea is that any real number $x$ will be $\sup A$ for some
Dedekind cut $A.$
The problem with this is that in order to write $x = \sup A,$
we must be able to evaluate $\sup A,$ that is, we need to identify a number that satisfies the necessary conditions to be $\sup A.$
But since we are still in the process of defining the real numbers,
the only numbers we have available to use as values are numbers such as
the integers or the rational numbers, and $\sup A$ is not yet defined for most of the Dedekind cuts.
So in order to initially define the real numbers so that we can start working with them, we take a Dedekind cut $A$ itself as the definition of a real number.
The intuition is still that the real number we define in this way is (or should be) a supremum of the cut $A,$ but in order to avoid a circular reference we avoid making that part of the definition.
Of course, once we have defined the real numbers (using Dedekind cuts or any other method), we can construct a set of real numbers $A'$ that corresponds exactly to any given Dedekind cut $A$ 
(that is, the real number $r$ is in $A'$ if and only if $r$ corresponds to one of the rational numbers in $A$),
and then we expect to find that $A = \sup A'.$
In other words, once the real numbers have been defined,
each real number is the supremum of its own Dedekind cut.
A: The Dedekind cut splits $\mathbb Q$ in two subsets of rationals, all the ones smaller than the desired real, and all the ones larger.
These infinite subsets are used because a real might not be a rational, but can be approached arbitrarily closely by rationals. And by using infinitely many rationals, you can get closer and closer. (You need them all because there is no "closest" rational.)
For instance,
$$1<\frac{14}{10}<\frac{141}{100}<\frac{1414}{1000}<\frac{14142}{10000}\cdots<\sqrt2<\cdots<\frac{14143}{10000}<\frac{1415}{1000}<\frac{142}{100}<\frac{15}{10}<2$$
As the concept of reals can only be defined using already established concepts, the real is defined to be one of these sets of rationals.

If this approach seems contrived to you, remember that a rational is an infinite set of integer pairs $(kp,kq)$ where $p,q$ are relative primes.
From this definition, the basic operations (addition, multiplication, comparison...) on reals can be defined, by reasoning on the infinite subsets. But once the algebraic properties of these numbers are established, they can be manipulated as if they were "atomic" entities.
A: I concur with Nagase's answer, but I wish to elaborate more on the logical aspects. In short, reals are not entities we pluck out of thin air or fancy alone, but are rather described as a single whole structure that we believe is meaningful. So we can and should separate between the properties of such structures (Dedekind-complete ordered fields) and the question of whether they exist.
One can do practically all practical real analysis using the second-order axiomatization of the reals, but one should also separately show that such fields exist, so that the theorems about real fields are actually saying something!
To prove existence, one can constructed the set $R$ of all equivalence classes of Cauchy sequences of rationals, where two sequences are equivalent iff their difference tends to zero, then define $+,·,<$ on the classes, and then prove that the resulting structure $(R,+,·,<)$ is indeed a Dedekind-complete ordered field.
One can alternatively construct the set $R$ of all equivalence classes of Dedekind cuts of rationals (where two cuts are equivalent iff ...), then define $+,·,<$ on them, and then prove that the resulting structure $(R,+,·,<)$ is indeed a Dedekind-complete ordered field.
In either case, all theorems of real analysis that we prove using only the axiomatization of the reals will apply to $(R,+,·,<)$. Furthermore, we can show that the rationals as an ordered field embeds into $(R,+,·,<)$. Using this embedding we can construct another structure $(R',+',·',<')$ which is a Dedekind-complete ordered field that contains the rationals directly. It is precisely because of this that we usually consider the rationals to be a substructure of the reals. This is the actual source of your confusion.
In fact, one can prove that the second-order axiomatization of the reals is categorical, meaning that it describes a unique structure up to isomorphism. In other words, any two Dedekind-complete ordered fields are isomorphic. This is the reason we usually call the reals "the reals", because there essentially is only one such structure (abstractly speaking).
Note that categoricity of the second-order axiomatization of the reals crucially depends on the second-order completeness axiom. See this post for some details on this.
A: To cut directly to the chase:

The definition states in the beginning "A real number is a cut...", but then it also states "A cut is a set of rational numbers..." So a real number is 'a set of rational numbers'?!

The word "is" here is a shortcut for a bijection. To be very exact, the book and your translation should read "there is a bijection between the set of the real numbers, and the set of cuts"; and further "there is a bijection between the set of cuts and the set of sets of rational numbers".
Hence, by transivity, you get "there is a bijection between the set of real numbers and the set of sets of rational numbers".
Obviously, this is a bit unwieldy to read and write, so "is" is used instead.
A: Have you seen the construction of the integers (from the natural numbers)? The integers are constructed as classes of equivalences of ordered pairs, which is also "weird". In order for you to begin to understand this process, start by thinking about those definitions as implementations, or models, of structures that we will (artificially, you can say, but that is irrelevant) show that behave like we expect them to behave in order for them to be called as such (integers, real numbers etc). Later you will see that this distinction is mostly psychological.
The bottom line is: They are clever ways to show the existence of objects which realize the structure we are idealizing. Dedekind cuts is a particularly clever example, as I'm sure you will eventually appreciate.
One way to start appreciating the cleverness behind this construction (and also dispel the negative feeling of artificiality and/or confusion) is by trying to define the real numbers by yourself. Be critical in such construction, and you will realize that a lot of your attempts will be (most likely) circular. 
A: The secret is not to think too hard. Take a real number x, and take all rational numbers q < x. Now, the set of all rational numbers q < x satisfies the definition of a cut. In the same fashion, given a cut according to the definition of set of rational numbers you describe, you can find a real number x which is the sup of that cut.
So, in a way, you can identify a cut (a set of rational numbers) with a real number.
A: I haven't got time to do a lot of this right now, but I googled "eudoxus theory of proportions" and one of the first things to come up was this.  Note in particular on page 53:

Eudoxus’ idea was to say that a length... is determined by those rational lengths less than it and those rational lengths greater than it.

In other words, this is essentially Dedekind cuts, 2300 years before Dedekind!
IMHO a bit of historical context often helps with this sort of thing.  Try googling for yourself, you may find something which helps you even more than the above link.
A: I think your problem with Dedekind's definition is rather philosophical. You understand the definition, you just don't like it. If it helps, let us look at another definition for a real number which might be more intuitive. One given by Cauchy (Egreg has pointed out that this definition is due to Cantor):
A real number is a class of equivalent Cauchy sequences with terms in $\mathbb{Q}$. Yes, unfortunately we still can't get rid of the idea of representing a real number by a set!
Two Cauchy sequences are equivalent iff the difference between them goes to $0$. The idea is simple. You know that a real number has a decimal expansion. For example:
$$\sqrt{2} = 1.4142135623730950488$$
So, you can define a sequence of rational numbers that converges to $\sqrt{2}$:. $$a_1=1,a_2=1.4,a_3=1.41,a_4=1.4142$$ and so on.
The thing is that this sequence of rational numbers is converging to a number that is not rational. This shows that rational numbers have holes! 
But what is so good about a Cauchy sequence? A Cauchy sequence is a sequence where its terms are getting closer and closer and closer to each other. So, we expect it to converge to something if there is no "hole" in our space. If there is a hole, we can't converge to it. But we can manually/artificially add it to our original space and "complete" our space. This is how real numbers are obtained from rational numbers in real analysis: as the completion of the rational numbers with the Euclidean metric.
A: You will find that all sorts of things which seem obvious are in fact defined as sets - one aspect of this is the way that the logical foundations of mathematics are often with reference to set theory. 
In fact, if you build mathematics from set theory, you find that natural numbers are defined as sets, and these are generalised to ordinal numbers, which include infinite sets.
Then when we define the rational numbers we want $\frac 12=\frac 24=\frac 36=\dots$ and one way of doing this is to define a rational number as an equivalence class (a set) of ordered pairs of integers.
We recover the normal way of looking at things by giving these sets names (so we name the numbers we use), and we tend to forget the underlying structure once we have checked that everything is well-defined.
The importance of the Dedekind construction using cuts is that it constructs a collection of numbers which have the properties we expect and hope and want. We can in fact show that (given the correct definition) any collection of objects which has the properties of the real numbers is isomorphic to the reals - it has essentially the same structure and the same properties and there are no surprises. But we still need to know that such a collection of objects exists - and that is what Dedekind shows. Once we have these properties with uniqueness and existence, we can, in fact, choose our favourite names for the objects with confidence.
A: The OP's translated definition is somewhat deficient. The idea of a "cut" really deserves its full definition before you can leverage that to construct the "reals". The fact that the OP's quoted definition intermixes the two is what personally gives me grief; and raises the circular-definition issue in this related SE question. Compare to the short definition at Wikipedia:

A Dedekind cut in an ordered field is a partition of it, (A, B), such
  that A is nonempty and closed downwards, B is nonempty and closed
  upwards, and A contains no greatest element. Real numbers can be
  constructed as Dedekind cuts of rational numbers.

Note that the first sentence defines a "cut"; the second sentence suggests how to define a "real". (More detail is at the Wikipedia link, of course.)
Conceptually, we would like to give a proper definition to real numbers, but of course we can't use the term "real" or else have a circular, not well-defined term. So we can only refer to more primitive concepts, such as sets and rationals. The basic intuition is that if the number line were continuous, any "place" (i.e., "cut") that separates it into two parts would in fact be a usable number. Since such a place-cut may not be rational, we are forced instead to formally talk about the set of all rationals less than that place-cut. 
As other answers have noted, this is a consistent practice in formally defining other sets of numbers. Natural numbers are defined as sets of different cardinalities. Integers are connoted as sets of differences of naturals. Rationals are equated to sets of quotients of integers. And so forth. 
A: The question posed by the OP has been adequately answered here, but let me give a bit of background that might put the ideas in context. 
First, (one of) the other definitions of the Real Numbers is that a real is an Equivalence Class of Cauchy Sequences of rationals. Both the idea of a Cauchy Sequence (infinite sequences whose terms are "eventually" close together), and the idea of an Equivalence Class (things "related" to each other in an Equivalence Relation). This, like Dedekind Cuts is a quite beautiful thing, but also difficult to grasp immediately. It is an abstraction built up of other abstractions which are themselves... recursively back to (perhaps) the Peano Axioms. 
But the reason for this note is the following. 
Mathematics of Number is built from two things, Sets of Axioms and Definitions. The Peano Axioms don't "define" the notion of 0 (I'll start with 0 rather than 1 to simplify the argument here), except in the context of the other axioms. Think of the Peano Axioms as a "collective definition" of the Natural Numbers. 
However, to get "interesting things" to talk about, mathematicians also define things based on the axioms (and earlier definitions). So we can define Prime Numbers after a bit of work. If you look, most of the theorems of mathematics talk about things defined, not the axioms per se. 
But we can define The Integers as equivalence classes of ordered pairs of natural numbers. So -3 (in the usual notation) is the equivalence class that contains (2, 5) whereas 3 (as a member of the integers) is now the class that contains, for example (7, 4). The notion of "negative" is derived from this, rather than part of the definition. And the simpler concept (Natural Number) embeds naturally into the new one (Integer).  
Likewise we can define Rationals based on simpler things (Integers), and eventually Reals based on those. 
These definitions give us "interesting things" to study, resulting in Theorems that specify how these new things behave. 
But the crux is this. What is the alternative? 
We could, of course, though with some difficulty, simply abandon the Peano Axioms and create a new set of axioms for the Integers and then yet another set for the Rationals, etc. This seems appealing initially, but leaves us with the problem of proving that the various axiom systems are consistent with each other. That can be very difficult, and mathematicians have, in general, chosen the axiom + definition path as the preferred one. 
That isn't to say that new axioms are never introduced - The Axiom of Choice, for example. 
Back to Dedekind Cuts for a moment. Note that when you give two definitions of something, some Theorems will be easier with one of the definitions and others will be harder. That is expected. But even here you need at some point to show that the two definitions are consistent with each other: that the notion of Real as "Dedekind Cut" and the notion of Real as "Equivalence class of Cauchy Sequences" are the same notion. 
Only then can you say that $v = sqrt(2)$ is both the set of rationals whose square is less than 2 and the set (equivalence class) of Cauchy sequences that "converge" to v. 

Note the definitions here are informal, inviting further study. Also to avoid making this a textbook. In particular, I left "convergence" undefined. Nor did I say which equivalence relation of Cauchy Sequences is used to define Reals. That is all intentional. 
A: A Dedekind cut is most commonly described as a partition of the set of rationals $\mathbb Q$ into two non-empty sets $A$ and $B$, such that $\forall a\in A, b\in B\ (a<b)$. And that, IMHO, is more comprehensible. See Wikipedia Dedekind cut.
The definition in your textbook just uses a half of the partition, the set $A$ – but it is equivalent to the one above, because the properties defined clearly imply the $B$ set.
A: The OP should have no trouble understanding the following mathematical investigation of 'cuts' by clearing their minds and starting fresh; the material is presented as a warm-up / motivational exercise.
Before starting, recall that if $q \in \mathbb Q$ and $0 \lt q \lt 1$ then there are positive integers $d$, $n$, and $m$ so that algebraically
$\tag 0 q = \frac{n}{d} \text{ and } \frac{n}{d} + \frac{m}{d} = 1$
and that we can look at this as breaking unity into two complementary pieces. We have $n$ parts on the left side and $m$ parts on the right side. We can generalize this idea of proportion shown in $\text{(0)}$.
Let $\mathcal U = \{q \in \mathbb Q \; | \; 0 \lt q \lt 1\}$.
Let $A \subset \mathcal U$. We call $A$ a left chunkette of $\mathcal U$ if it satisfies the following:
$\tag 1 A \ne \emptyset$
$\tag 2 \text{If } a_1 \in A \text{ and } a_0 \lt a_1 \text{ Then } a_0 \in A$
$\tag 3 A \text{ has no greatest number}$
In a similar manner, we can define a right chunkette of $\mathcal U$.
We define a cut $\gamma$ of $\mathcal U$ to be an ordered pair $(A,B)$ of subsets $A, B \subset \mathcal U$ satisfying the following:
$\tag 4 A \cap B = \emptyset$
$\tag 5 A \text{ is a left chunkette and } B  \text{ is a right chunkette}$
$\tag 6 \text{For every integer } n \gt 0 \text{ there exist } a \in A, b \in B \text{ such that } b - a \lt 1/n$
Let $\hat{\mathcal U}$ be the collection of all cuts of $\mathcal U$.
Exercise: Define a (natural) injective mapping of $\mathcal U$ into $\hat{\mathcal U}$.
The interested reader is invited to extend these ideas by defining cuts (as 'kissing' chunkettes) on $\mathbb Q$ and then creating the real numbers.
A: I'd like to give a programming perspective (or really, an analogy) over here.
Suppose you want a stack, a data structure, which basically allows access only to the "top of the stack" in terms of pushing or popping elements on the top, but never anywhere else:

However, you are working with a (relatively) low level programming language, which is only equipped with memory operations such as allocating some sized memory block or freeing it up. But you can still implement a stack.

The key word here is implement. Although your programming language did not "have" stacks, you could still make a satisfactory implementation of stacks that simulate a stack.
How do you know that you have implemented the stack correctly? When you cannot tell the difference between a "real" stack and the simulated stack by interacting with it through the methods exposed.
To draw the parallel, what we are doing in mathematics is: starting with some machinery like sets, natural numbers, integers, real numbers and pretending that real numbers are not available in our language. Then we ask, can we somehow "implement" real numbers using the machinery we already have? Again, what does it mean for an implementation (or construction) of real numbers to be correct? Well, it should follow the real number axioms.
This is an ubiquitous themes in the Foundations of Mathematics. Another interesting example is the implementation of natural numbers in lambda calculus.

So, the question now is, why not just start with real numbers built-in to your language? By that, what I mean is you could extend your formal language to say that "there exists a complete ordered field...". The problem is that, how do you know that adding this axiom is still a sane thing to do? Sure, for real numbers, it might feel intuitive, but as a philosophical endeavour, we would like to see if it can be constructed.
For what it is worth, there are other ways to construct real numbers other than dedekind cuts, and all of them are equally correct. However, real analysts do not reason about real numbers pretending that they are a cut, or that they are a cauchy sequence in their daily work since the axioms are a more "high level interface" to work with.
