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'.