Is the concept of a uniform space elementary? I'm self studying with Munkres's topology and he uses the uniform metric several times throughout the text. When I looked in Wikipedia I found that there's this concept of a uniform space.
I'd like to know what are it's uses (outside point set topology) and whether it's an important thing to learn on a first run on topology? 
 A: The concept of uniform space is designed to abstract the notions
of uniform continuity, Cauchy sequences, and completeness from
metric space theory. 
What these concepts all have in common is that they require us to
be able to talk about a pair of points $x$ and $y$ being "sufficiently
close", where the notion of "sufficiently close" doesn't make explicit
reference to either $x$ or $y$.
In an arbitrary topological space, we can't do this: we can use the
system of n.h.s around a fixed point $x$ to measure "closeness to $x$",
but we can't make sense of the statement "$y$ is as close to $x$ as $y'$ is to $x'$" (if $x$ and $x'$ are different points) because there is no mechanism for comparing the size of a n.h. of $x$ to the size of a n.h. of $x'$.
In a metric space, we can do this, because we can talk about the $\epsilon$ n.h.s of any point, for fixed values of $\epsilon$.
In a topological group, we can also do this, because we can use translation by group elements to compare the system of n.h.s around any point to the system of n.h.s around some chosen base-point (usually the identity).
As Daniel Fischer indicates in his comment, the concept of uniform space
provides a language which generalizes these two examples.  I did find it useful to learn, since it clarifies results such as the theorem "closed and bounded implies compact" for subsets of Euclidean space, and since it provides a useful background language when studying topological vector spaces (and these sometimes come up in my research).  However, it is not used much in other areas of mathematics (in my experience), e.g. essentially not at all in algebraic topology or differenial topology, and so I wouldn't say there is any real need to learn it unless you feel highly motivated to do so.
A: Uniform spaces are right on the boundary of formalisms that are worth knowing.  Topological groups are definitely worth knowing, and are central to analysis and its applications.  The topology of a topological group is determined by the system of neighborhoods of the identity.  They also have a natural notion of uniform continuity.
In applications, the topology on a group is frequently given by a family of pseudometrics.  (An simple example is a Banach space -- a vector space is a kind of group, and the topology on the group is given by a single metric.)  Families of pseudometrics also have a notion of uniform continuity.
The axioms of a uniform space generalize these two cases — except that it's not a generalization.  The topology on a group always given by family of pseudometrics.  The interesting thing is that this proof — which can be phrased in terms of neighborhoods of the identity — immediately generalizes to all uniform spaces.  So while uniform spaces are not any more general, once you do the work of proving that theorem for topological groups, you get uniform spaces for free.
So if you don't like the definition of uniform spaces, or find it hard to understand, the idea is not strictly necessary.  But it's not much more work beyond understanding neighborhoods of the identity of a topological group.
A: Let me quote from Warren Page's Topological Uniform Structures:

This book aims to acquaint the reader with a slice of mathematics that is interesting, meaningful, and in the mainstream of contemporary [Ed: book originally published 1978] mathematical developments. Admittedly a number of excellent sources cover, in part, uniform spaces, topological groups, topological vector spaces, topological algebras, and abstract harmonic analysis. 

and 

The overall unifying theme of topologies compatible with increasingly enriched algebraic structures ... a number of striking results that combine and interlace algebraic, topological, and measure-theoretic properties associated with the structure under consideration. 

As indicated, the subjects which most directly benefit from some background in uniform spaces are


*

*topological groups

*topological vector spaces

*topological algebras

*abstract harmonic analysis


But the uniform structure often only becomes apparent when you get quite far in the study of these objects. For example, while there is a rich and interesting theory of topological vector spaces, most of the TVS that are used commonly in other fields (in particular Banach and Frechet spaces) are in fact metrizable, so intuitions from metric spaces are "good enough" for everyday use for many mathematicians. 
Let me give another example since you mentioned differential topology in your comments: any manifold is locally Euclidean, and hence locally metrizable. It is a theorem that locally metrizable topological spaces are metrizable if and only if it is Hausdorff and paracompact. When you study differential topology most of the time Hausdorff and paracompact are built-in assumptions for your manifolds. Hence for the most part, the study of smooth manifolds can be dispensed with using intuitions built up from metric spaces, without necessarily having to delve into intricacies associated with uniform structures. 
On a first run through topology, I think it is safe to put-off learning about uniform spaces until later. By the time you really need it, you can probably pick it up relatively quickly. The one advantage to thinking a little bit about the uniform spaces (especially how they differ from metric spaces) is that it forces you to confront certain intuitive prejudices that we've grown accustomed to from working with $\mathbb{R}$ all the time, and allows you to overcome certain limitations that arises from thinking only about countable, instead of uncountable infinities. (This of course comes up also in the difference between nets and sequences.) 
