What does a topology do, and what makes a particular topology the 'right' one? From Wikipedia:

The same set can have different topologies. For instance, the real line, the complex plane, and the Cantor set can be thought of as the same set with different topologies.

[Aside: Taken literally, this claim is either subjective (hinging on the interpretation of "can be thought of as") or false. I would appreciate it if someone could clarify what this is meant to say.]
From so many sources that it's a cliche:

The Euclidean topology is the natural topology for $\Bbb{R}^n$...
Naturally, sets such as $\Bbb{N}$ and $\Bbb{Z}$ [which are not dense / nowhere dense] suggest the discrete topology...


Both of these sentiments seem common enough, and suggest to me that particular topologies are implicitly descriptive of particular sets and vice-versa. Yet, there seems to be no grounds for "$X$ topology 'naturally' describes $Y$", save for $X$ and $Y$ being nominally related (in the same manner as "ocean" and "giant tube worm").
In fact, it seems to me that the choice of topology is fundamentally arbitrary. In a discipline that prides itself on rigor, I find this, along with the casual acceptance of very loose explanations as to why a particular topology is 'natural' or 'obvious', slightly disturbing.
All of this has led me to two very important question:

*

*What precisely does a topology do - that is, given a particular topology, what can you infer about the structure (i.e. 'shape'), of a space with that topology (aside from the obvious 'this is open/closed', 'this is continuous', etc)?


*What, if anything, makes a specific topology the right one for a given context - that is, given an arbitrary set, and perhaps some additional information, which topology best describes that set, and why?
Ideally, when presented any problem I want to be able to answer the question "which topology should I use for this"?

Note: I am aware of a what a metric topology is and how it is used, I am not asking for an explanation of metric topologies, I want to know why the selection of any particular topology is reasonable in the first place. Seeing as how a metric is just one of any number of characteristics which can be assigned to a set arbitrarily, I would like to know why any one characteristic should be selected to define the 'natural' topology on a set.
Edit:
Clarification
I did not expect this question to get as much of a response as it did, but after reading the comments, I think some clarification is in order.
'Arbitrary' means subject to individual choice, nothing more, nothing less. Because the topology assigned to a set is independent (up to inclusion) of that set, the choice of topology is indeed 'arbitrary'.
That being said, I can understand why the Euclidean Topology in $\Bbb{R}^n$ (particularly considered as a vector space) would seem 'natural' - it follows what we would expect from our intuition. It is tempting to say that this intuition is 'natural' or even 'universal' - an element of human nature rather than a product of a particular construct - however, if cultural anthropology has taught me one thing, it is that nothing is universal (except for homicidal endocannibalism taboos). The intuition behind a particular understanding of 'space' is something learned - it is not known from birth, nor is it present in all people.
More generally, topology is not an implicitly spatial thing. Outside of analysis and geometric topology, a topology is just a kind of set.
My Take Away
From what I can piece together from the answers, a topology first and foremost is a means of labeling particular subsets of a set - namely those subsets possessing a desired property.
The significance of the labelling depends on what information we are interested in, not the characteristics of the set itself. In the case of metric topology, what we are interested in is 'space', and the open sets of a metric topology effectively captures the idea of 'sets that take up space'.
In other cases, such as Furstenberg's proof that there are infinitely many primes (addressed in J.G.'s answer) or algebraic geometry (addressed in Moishe Kohan's answer) we are interested in different things - like 'sets which are infinite', or 'sets containing the zeros of polynomials'.
 A: 
What does a topology do, and what makes a particular topology the 'right' one?

A topology lets you prove certain things; it's the right one if it helps you with your present problem; it's the natural one if it helps you 99 times out of 100.
To illustrate this, I'll give an example of a problem you'd never think is solvable with topology, but it absolutely is. This is Fürstenberg's proof there are infinitely many primes (with some steps skipped you can try filling in yourself):

We can verify there is a topology on $\Bbb Z$ whose open sets are the
  unions of two-sided arithmetic progressions in $\Bbb Z$. In this
  topology, each open set is also closed, and is either infinite or
  empty. If there are finitely many primes, the set of integers not divisible by any prime
  number is clopen, and hence infinite or empty. But this set is
  $\{-1,\,1\}$, giving a contradiction.

This example inspired this question; all its answers are worth reading for other "creative" topology examples. (My own answer there discusses another example that proves any Jordan curve passes through some rectangle's vertices.) But that question is about unexpected uses of topology. The proofs' authors always picked the right topology, but typically not a natural one.
Topologies are intimately related to metrics, norms etc., the choice of which may hint at what is natural. For example, our options on $\Bbb Q$ are limited, and the Euclidean norm gives birth to $\Bbb R$ and $\Bbb C$ in a way that non-trivial alternatives give us the $p$-adics, which continue to have limited intuition and utility.
A: Let me start with the sentence 

The same set can have different topologies. For instance, the real line, the complex plane, and the Cantor set can be thought of as the same set with different topologies.

This is totally correct and, at the same time, useless. To make sense of this statement, one observes that all the three sets have the same cardinality. Hence, one takes, for instance, a bijection $f: {\mathbb R}\to {\mathbb C}$ and takes the image under $f$ of the standard topology ${\mathcal T}_1$ on ${\mathbb R}$. The result is a topology ${\mathcal T}_2$ on ${\mathbb C}$ such that  $({\mathbb C},{\mathcal T}_2)$ is homeomorphic to $({\mathbb R},{\mathcal T}_1)$. However, given the fact that our map $f$ was an arbitrary bijection, we did not gain any insight into topological structure of either one of these spaces. 
Next, to the 

The Euclidean topology is the natural topology for ${\mathbb R}^n$.

the answer is "it depends". First of all  you have to decide what ${\mathbb R}^n$ is. I assume this means the $n$-dimensional real vector space (with a particular choice of a basis). If you are doing, say, analysis or PDEs or differential geometry or geometric topology on this space then indeed, the Euclidean topology is the natural choice. For instance, from the analysis viewpoint, this topology is natural since it agrees with the notion of limits, continuity etc. that one uses in analysis on ${\mathbb R}^n$. In fact, (real) analysis was one of the original sources of topology which arouse out of needs of several branches of mathematics and analysis (real and complex) was one of these.
At the same time, if you are doing (real) algebraic geometry, you quickly realize that besides the Euclidean topology, there is another natural topology on  ${\mathbb R}^n$, namely, the Zariski topology (which is non-Hausdorff as long as $n>0$). This topology is strictly weaker than the Euclidean topology but captures solutions of algebraic equations better than the Euclidean topology. The same (even more so) applies to ${\mathbb C}^n$. Moreover, in algebraic geometry one frequently uses both topologies and compares the answers. Furthermore, the Zariski topology on ${\mathbb C}^n$ has a mild generalization, called the etale topology (which is, strictly speaking not a topology on on ${\mathbb C}^n$) which is also quite useful. So, from the algebraic viewpoint, say, Zariski topology is more natural than the Euclidean topology. 
One way to think of this is that Zariski closed sets are zero sets of polynomial (vector) functions while subsets of ${\mathbb C}^n$ closed in the classical topology are zero level sets of smooth functions. 
Dealing with topology such as the Zariski topology (on general algebraic varieties/schemes) forces one to reconsider the standard topological notions one is used to in analysis, such as compactness and connectivity. One replaces them with appropriate "algebraic" counterparts (completeness and irreducibility).   
Thirdly,

What, if anything, makes a specific topology the right one for a given context - that is, given an arbitrary set, and perhaps some additional information, which topology best describes that set, and why?

does not have a universal answer. As you learn more mathematics, you will discover this from reading and trying to solve problems. If your math specialty is, say, PDEs, then you use Euclidean topology on ${\mathbb R}^n$ and "whatever works" on various functional spaces. For instance, you try to gain some form of compactness (if possible), which requires "fewer" open sets and continuity of some functionals (which requires more open sets). There is a clear tension between the two, which makes things interesting. 
Edit.  A classical example (going back to the origins of topology) is the Dirichlet problem. The idea (due to Riemann) is to solve the problem by minimizing a suitable (nonlinear!) functional (the energy) on a certain space of functions. Thus, the topology one looks for is the one for which the energy functional is continuous and its sublevel sets are compact. Riemann's original expectation was that even in this infinite-dimensional setting, a continuous functional on a closed bounded subset will attain its minimum (the "Dirichlet principle"). Hilbert observed that this is utterly wrong in full generality, but can be made work in the special setting of the Dirichlet problem by choosing carefully the functional space and its topology.  
One more example, this time from algebra: Topology on groups. In general, in algebra it is a good idea to use topology “consistent with the algebraic structure as much as possible”. For instance, for the group $G=SL(n, {\mathbb C})$ the Euclidean (also called “classical”) topology will equip $G$ with the structure of a topological group (multiplication $G\times G\to G$ and the inversion $G\to G$ are continuous) and even of a Lie group (the group operations are differentiable with respect to the natural smooth manifold structure). However, in many cases, it is also useful to equip the same group $G$ with the Zariski topology. And with this topology, $G$ is only a paratopological group: The multiplication $G\times G\to G$ is continuous with respect to each “variable” but is not jointly continuous. But looking at the situation a bit closer, the source of the problem is that we are using the "wrong topology" on the product group $G\times G$: Instead of the product topology one should use the Zariski topology on this product. 
In this example you see how algebra (both group theory and algebraic geometry) governs the choice of topology (on both $G$ and $G\times G$). 
The group $G$ with Zariski topology has much fewer closed subgroups which makes it possible to analyze them. Furthermore, they are frequently “nicer” than closed subgroups in the classical topology. The example that I like is taking the Zariski closure of an infinite subgroup $\Gamma$ which is discrete (and, hence, closed) in the classical topology. The resulting subgroup of $G$ is again a Lie group but now it has only finitely many connected components (in the classical topology!). Hence, this Zariski closure can be further analyzed by means of the Lie theory (say, looking at its Lie algebra). So, in this example, we go from   Euclidean topology to Zariski topology and then back to Euclidean topology. 
A: This answer won't be so much about topology as it will be about the notion of "natural" objects existing in mathematics in general. To begin, you are absolutely right. The notion of a natural topology is entirely subjective and there is no mathematical rigor behind it. But that does not mean it is not useful.
To explore this further, it makes sense to talk about definitions as a whole. While you seem to understand this, there have been quite a few definitions on this site asking why things are defined a certain way, and whether a definition is "right" or "wrong." While answers may differ, the underlying thread is consistent: Definitions are entirely a convenience, and cannot be right or wrong. Coming back to your example, this makes it clear that natural topologies are natural simply because most mathematics agree with this definition; most mathematics tend to study certain kinds of problems, and in these problems some topologies tend to naturally occur quite often, and so they will be referred to as natural. If you do work that is similar or related to what many others do, then you might too find this convenient. On the other hand, if you invent a whole new branch of mathematics in which other topologies (or really any object) are natural, you might choose to redefine natural to mean something else.
A: "real line, the complex plane, and the Cantor set can be thought of as the same set". Yes, and no. Yes, thanks to Cantor, relying on the existence of a bijection that 1. is usually difficult to describe precisely, and 2. does not respect the structure of these sets that we are most interested in (e.g. the the algebraic operations of real and complex number, total order of the reals, etc). So, no, because in order to say these are the same sets, you need to forget their pertinent structure, e.g. the topology of the real line could be defined entirely in terms of its order, but now you want to disregard the lineal order of the line. 
If a definition is simple (yet of some use), then it is natural. The discrete topology has a simple definition, so it is natural. Sometimes the discrete topology seems useless, but for example the Cantor set could be thought of the product of countably many copies of the discrete doubleton $\{0,1\}$, so suddenly the discrete topology becomes related to the standard topology on the Cantor set induced from the real line. Interrelations like this between two topologies improve our understanding of the objects we study, and both topologies gain from such interplay, and the more examples we see along these lines, the more these topologies would seem natural. 
True, for the real line (and $\Bbb R^n$) the term natural topology means the Euclidean, or metric topology. The definition of topology via open balls (or open intervals) is 1. easy (enough) and 2. useful (in formalizing the notion of continuity, which was first conceived of, I would think, before the invention and formalization of the notion of topology). The fact that 
the order topology and the metric topology of the real line are the same says there ought to be something natural about this topology. Similarly, the metric topology on $\Bbb R^n$ and the product topology coincide, and it ought to be of some significance that we arrive to the same topology via different means. Taking away the structure that is present in the objects that we study, and stripping these objects 
(real line, complex plane, etc) from most of the features that make them interesting to us, to study, just to say oh, they are the same set ... this is just too simplistic and really doesn't say anything. Set theory is an interesting subject to study on its own, but saying all spaces are "the same" just because they are the same set, well, on one hand it is abuse of set-theoretic notions taken out of context, and on the other hand doesn't really say much, given that the bijections involved rarely preserve the structures that make these spaces interesting to study. (For example, again, we could use the total order of the real line and a bijection to the complex plane, to define a total order on the plane, but then we wouldn't have much use of it, since 1. we would not have a simple enough description of the bijection, and 2. a total order on the complex plane would not play very well along with the usual structure of the complex numbers.) 
It is not clear if I am answering your question (but the question itself is broad and might be subject to interpretation), I am not saying that there is not a nice interplay between set theory and topology. I just read an answer showing there is no real-valued function $f$ on the reals having limit $\infty$ at every point. 
Indeed if $A_n = \{x \in [0,1]: |f(x)| \leq n\}$ and if $A_n$ is infinite for some $n$ (involving set theory here), then $A_n$ must contain a converging sequence $x_k$, and we cannot have $\lim_{x_k\rightarrow x}f(x_k)=\infty$. But, we cannot have all $A_n$ being finite either (for each $n$), since then the interval $[0,1]$ would be countable, a contradiction (and again, 
use of set theory). But the use of set theory here gives us something, whereas if you define a topology on the complex plane by simply transferring the topology of the real line via the use of a bijection, onto the complex plane, I do not see much use of this "transferred" topology, it won't play well with the structure of the complex plane, and if it doesn't, then it won't be natural. 
So, I guess, a starting definition for a natular topology would be a topology that plays well with some of the structure that you already find useful and interesting on the set that you consider (and perhaps helps you solve some problem, as in the Furstenberg's proof of the infinitude of the prime numbers, discussed in another answer, and many other examples, some also given in another answer). 
So, again, when you say "the same set with a different topology" then you totally ignore whatever other structure you might be interested in (order, algebraic operations, etc). If you ignore the other structure, then, it becomes next to impossible to define which topology would be natural. Perhaps you could say that the discrete topology on an arbitrary set is more natural than the indiscrete topology (since if you take a product of discrete spaces you do get a topology of interest), but I don't see how one would go very far in an attempt to define a natural topology on a set, without taking into account some other interesting or existing structure on the set in question. 
