Interpretation of a real numbers through Dedekind's cuts For what I have understood Dedekind, in order to define an irrational numbers such as  $\sqrt{2}$ defines $2$ sections in the number line, one with all the rational numbers that squared are greater than $2$ and one with all the rational numbers that squared are less than $2$. Now Dedekind says that the section is an irrational number ($\sqrt{2}$). But what does that mean? The only way I could interpret it is that $\sqrt{2}$ is the number on the real line which is excluded by the $2$ cuts. Is this correct? What does it mean for a number to be a section? And if my interpretation is correct then Dedekind hasn't really demonstrated the existence of  $\sqrt{2}$ but has merely said that it is there. Do I need to consider the fact that the real line is continuous?
 A: If we want to define irrational numbers, that means we are assuming we are only starting with rationals. Its not just that we don't know what $\sqrt{2}$ is, its that it doesn't exist yet. That means we have to actually define what that means. 
Why do we want to do this? Well it has something to do with this fact that the real line is 'continuous' as you say. The problem with the rationals is that they aren't what mathematicians would call complete - this is essentially what you mean by 'continuous'. An example of this incompleteness is that we can find a sequence of rational numbers that 'converge' to $\sqrt{2}$ (although this is a strange notion, since $\sqrt{2}$ doesn't exist yet). An example would be a sequence defined by the decimal expansion of $\sqrt{2}$: $1,1.4,1.41,1.414,1.4142,\dots$. 
Okay, so we've realized the rational numbers aren't quite what we want, so we need some new numbers. How do we create numbers? Well, you may recall that we can define the natural numbers by the Peano axioms, or that we constructed the rational numbers as equivalences classes of ordered pairs of integers (more specifically as partitions of $\mathbb{Z}\times\mathbb{Z}\setminus\{0\}$). Notice that with the rationals, our 'numbers' are actually sets of other numbers we previously defined (in particular, $(a,b)$ and $(c,d)$ are part of the same rational number if $ad=bc$ - usually we would write this as $\dfrac{a}{b}=\dfrac{c}{d}$). 
When we define the real numbers (using Dedekind's definition), we will use a similar idea. These 'sections' are sets of rational numbers. These are in fact the real numbers. Later, when we use them in calculus or other math, we will just think of them as points on a number line, but in a rigorous definition, they are in fact sets. 
Now, just to clarify, the two sets for $\sqrt{2}$ are given by $$A=\{r\in\mathbb{Q}:r\le 0\}\cup\{r\in\mathbb{Q}:r>0\text{ and }r^2<2\}$$ and $$B=\{r\in\mathbb{Q}:r>0\text{ and }r^2>2\}$$ where $A$ is the lower section and $B$ is the upper section. (Note this is slightly different than what you mentioned. The way you stated it, $-4$ would be in $B$ not $A$.) In particular, these sets 'coincide' with $$A'=\mathbb{Q}\cap\{x\in\mathbb{R}:x<\sqrt{2}\}$$ and $$B'=\mathbb{Q}\cap\{x\in\mathbb{R}:x>\sqrt{2}\}$$ I say 'coincide' in quotes, since we are viewing $A$ and $B$ as subsets of $\mathbb{Q}$ and $A'$ and $B'$ as subsets of $\mathbb{R}$. This distinction is important, because as we discussed above, $\mathbb{R}$ is a subset of the power set of $\mathbb{Q}$. 
To summarize, you have the right idea in your head. The real number we are constructing with these sections is the 'missing' one. However, formally, these numbers are just sets, not points on a number line (and be careful about calling it the real line - these cuts are from the rational line!). And yes, Dedekind has 'merely said it was there' - but really this is a big deal and this is what math is all about. There was a concept that was missing from our understanding, so Dedekind defined a new one that would be useful. 
A: Dedekind is defining the real numbers from scratch: imagine you understand what rationals are, but have no conception of real numbers. Dedekind tells you that you look at the collection of Dedekind cuts; he shows

*

*How to define arithmetic operations on these cuts.


*How to assign each rational number a cut.


*And why the collection of cuts, unlike the collection of rationals, is complete in a precise sense.
Dedekind then makes a really interesting decision: he decides that the collection of cuts is the "right" object to study, and he calls Dedekind cuts real numbers. For Dedekind, "$\sqrt{2}$" is shorthand for the pair $$\langle\{x: x^2<2\},\{x: x^2>2\}\rangle.$$ Whenever Dedekind uses the term "real number," he means "Dedekind cut."
This is a really odd approach to defining real numbers - we usually think about real numbers as points, not sets in any sense. But formalizing what a real number is is extremely difficult; Dedekind observed that using set theory, we can define a mathematical object that behaves exactly how our naive real numbers behave; he then (arguably in an act of bad ontology) proposed that these were literally the real numbers.
If you like, you can view Dedekind cuts as describing, rather than being, real numbers as you say: a given cut describes the unique real not in either of its sections (or the greatest element of its left section, if such exists - this happens when the cut describes a rational). This is much more philosophically conservative; but Dedekind adopted a more ambitious position. Like it or not (and there are certainly good reasons to dislike it - see e.g. this article by Benacerraf1, which focuses on natural numbers a la set theory rather than real numbers a la Dedekind, but is still relevant), that's what's going on here, and why Dedekind is using the word "is."
1Paul Benacerraf: What Numbers Could not Be.
The Philosophical Review, Vol. 74, No. 1 (Jan., 1965), pp. 47-73.
DOI:10.2307/2183530, JSTOR
A: I think what's missing in your understanding here stems from how number systems are built in (most mainstream) mathematics. Usually, a number system is defined based on the ZFC (or ZF) set theory, where every mathematical object that we deal with is a set. Even the natural numbers are considered sets. In fact, one of the most primitive ways of building the natural numbers involves defining,
\begin{align*}
0 &:= \lbrace \rbrace \\
1 &:= \lbrace 0 \rbrace = \lbrace \lbrace \rbrace \rbrace; \\
2 &:= \lbrace 0, 1 \rbrace = \lbrace \lbrace \rbrace, \lbrace \lbrace \rbrace \rbrace \rbrace;
\end{align*}
etc. We can then extend the natural numbers into the integers, by a few methods, but basically by using ordered pairs (which are also made with sets!) to form two copies of the positive natural numbers. The rationals are then defined by using an equivalence relation on $\mathbb{Z} \times \mathbb{Z}$, where $(a, b) \sim (c, d)$ if and only if $ad = bc$. But, importantly, everything comes back to sets!
Now, we're trying to define the real numbers. We can take it for granted (because it is outside the scope of the problem) that $\mathbb{Q}$ exists and is defined. Now we want to use $\mathbb{Q}$ to define something that matches our intuition about what $\mathbb{R}$ is supposed to be (for example, we want to make it a field for which $\mathbb{Q}$ is a subfield). But, as for which sets specifically make up $\mathbb{R}$, that's up to the definition we're using.
Dedekind cuts are Dedekind's way of defining $\mathbb{R}$. Dedekind defines $\mathbb{R}$ as the set of closed, ordered intervals $I$ that are bounded above and have the property that $\mathbb{Q} \setminus I$ is also an interval. The rationals correspond to the set of such intervals with maxima, for example, $I = (-\infty, 1/2] \cup \mathbb{Q}$. (Note, $I$ is not an interval in $\mathbb{R}$, but it is in $\mathbb{Q}$).
The set $I = \lbrace q \in \mathbb{Q} : q^2 \le 2 \rbrace$ is another interval in Dedekind's $\mathbb{R}$ (this does require some proof), but one can show that it has no maximum. It's an example of something in $\mathbb{R}$ that doesn't correspond to anything in $\mathbb{Q}$.
We can also define a multiplication operation on Dedekind's $\mathbb{R}$. If $I$ and $J$ are such intervals, then one can show that the set
$$I \cdot J := \lbrace ij : i, \in I, j \in J \rbrace$$
is also in $\mathbb{R}$. If we take $I$ to be the interval in the previous paragraph (the supposed square root of $2$), then we can show that
$$I \cdot I = \lbrace q \in \mathbb{Q} : q \le 2 \rbrace,$$
i.e. the interval corresponding to $2 \in \mathbb{Q}$. So, in Dedekind's $\mathbb{R}$, the number $2$ has a square root.
A: Without thinking about "real numbers" at first, consider a very special type of pairs of sets that can be formed from a "universe" $U$ that has a total ordering property. That is, there exists an operator "$<$" such that for every $u_1,u_2 \in U$ either $u_1$ and $u_2$ are the same element of $U$, or u$1<u_2$ or $u_2 < u_1$; and also, the $<$ operator is transitive in that $u_1 < u_2$ and $u_2 < u_3$ together imply $u_1 < u_3$.
The special type of pairs of sets $(A,B)$ are on in which $A\cup B = U$ and every element of $A$ is less than every element of $B$ and
there is no greatest element of $A$.  That is, if $a\in A$ then $\exists a'\in A: a'>a$.  Let's arbitrarily call such a special pair of sets a "Dedekind cut" on $U$.
As an example, on the universe of rational numbers, consider any proposition $P$ about a single number that has the inherent properties that if $q$ satisfies $P$ then all numbers greater than or equal to $q$ satisfies $P$, and also that there is no rational number $\bar{q}$ such that $\bar{q}$ satisfies $P$ but no greater rational does.  Then property $P$ defines a Dedekind cut, with $A$ being all numbers that satisfy $P$, and $B$ being all rational numbers that do not.
Now comes the critical step:  You can carefully define abstract operations that map pairs of Dedekind cuts to some other Dedekind cuts.  For example, working with cuts $c_1$ and $c_2$, you can say that $q$ is in the $A$ set of "$c_1 \oplus  c_2$" if and only if $q=q_1+q_2$ with   $q_1\in A(c_1)$ and $q_2\in A(c_2)$.
It takes some care to ensure that this property does define a Dedekind cut, but it is not too hard.
So far we have not said a word about "real numbers." But after we have defined the usual arithmetic operators $\oplus,\ominus,\otimes,\oslash$ on our cuts in the universe of rationals, we might notice that there is a one-to-one correspondence between our intuitive notion of real numbers, and these Dedekind cuts on the reals, with our set operations preserving all results of the usual $+,-,\times,/$ operators.
Now comes the step that you are having trouble with:  Since these special pairs of sets can be put into 1:1 correspondence with our itnuitive notion of "the real numbers,"
we discard our fuzzy intuitive notion and say that the real numbers are these pairs of sets (Dedekind cuts).  What have we gained here?  We have put the real numbers on a solid footing without having to introduce any new axioms about arithmetic.
