First a minor correction to n!kofeyn's post: There are non-homeomorphic spaces with isomorphic homology groups. It works the other way around, i.e. if two spaces are homeomorphic (or even homotopically equivalent) then their homology groups are isomorphic.
Groups, rings, etc. are just abstractions of things a mathematician deals with every day. For example, commutative rings can be thought of as a generalization of the set \mathbb{Z} of integers with its additive and multiplicative structure. One additional thing we have in \mathbb{Z} is unique factorization into primes; this is such a deep property that it's profitable to see how it generalizes to arbitrary commutative rings, where of course we would need an appropriate analogue of "prime numbers". (This was an important observation back when people were trying to prove Fermat's Last Theorem. Actually, a lot of classical algebraic notions owe their existence to FLT.) Well it turns out that many rings, even nice ones, don't have any sensible unique factorization into primes. From an advanced point of view, the ring \mathbb{Z} is special because its ideal class group is trivial. What does this mean? To each "nice" ring R (by which I mean at least a Dedekind domain) one can associate (by following some sort of procedure) a special group G(R) which measures the extent to which unique factorization fails in R. For example, it turns out that G(R) is the trivial group if and only if R is a unique factorization domain. The ideal class group is a basic object of algebraic number theory.
This idea of associating algebraic invariants to things is very common in modern mathematics. Classically, mathematicians would associate numerical invariants to objects in order to distinguish between them. Here's a stupid example: we know that a square is different from a triangle because they have a different number of sides. A less stupid example would be to note that the topological space (0,1) \cup (2,3) \cup (4,5) is not the same as (=homeomorphic to) the space (0,2) \cup (5,6) because the former has 3 connected components wheras the latter has only 2.
Here's a more interesting example. To distinguish between compact Riemann surfaces, 19th century topologists associated to each such surface an integer called its genus, which basically counts the number of holes in the surface. Thus the sphere has genus 0, a torus genus 1, a 2-holed torus genus 2 and so on. And it turns out that a compact Riemann surface is completely determined by its genus. More generally, we can distinguish between closed surfaces (=closed connected 2-dimensional manifolds) by looking at two numerical invariants: their orientability (say, +1 if orientable and -1 otherwise) and something called their Euler characteristic (an integer). More generally still, we can use Betti numbers to try to distinguish between spaces. The Betti number of a space is roughly the maximum number of times we can cut the space before splitting it into two disconnected pieces.
There are plenty of other numerical invariants we can attach to a topological space, and 19th century topologists were almost exclusively concerned with coming up with new ones. But then came along Poincaré with the inspired idea of attaching an algebraic invariant to a topological space. He essentially defined the fundamental group (a.k.a. Poincaré group, or first homotopy group) of a topological space. What this group does is count the number of holes in the space (yes, this again! And this is not the end of it. There are several invariants concerned with counting holes in spaces. The reason for this is because this is actually a difficult thing to do in general) by looking at how loops can be continuously shrunk to a point inside the space. For instance, every loop in the plane can be continuously shrunk to a point, whereas in the punctured plane the loops around the puncture point cannot be shrunk without leaving the space. This information is encoded in the fact that the fundamental group of the plane is trivial, whereas the fundamental group of the punctured plane is not (it is isomorphic to \mathbb{Z}).
And n!kofeyn already mentioned the homology groups. These things count the number of holes as well, but in a more sophisticated way. There are also cohomology groups, K-groups, etc. etc. Some of these things are not only groups, but also rings, modules, vector spaces, etc. (And not all of them count holes!)
OK, enough about algebraic topology. Since you mentioned analysis then let me give you an example of where groups show up there. First let me say there are plenty (and I mean plenty) of such examples. There's no reason why I've chosen this one. Presumably you've done some or at least heard of Fourier analysis. There one begins by studying periodic functions on a compact interval of the form [0,2pi]. What's really going on, though, is that you're studying functions on the unit circle. The unit circle is a group (think of it as the set \{e^{ix} : x \in \mathbb{R}\} with complex multiplication), and Fourier analysis takes full advantage of this fact. Indeed, the unit circle S^1 is an example of what's called a topological group, which means a group with a topology which respects its group structure (i.e. the group operations of inversion and multiplication are continuous); in fact the topology is locally compact and the group is abelian, making S^1 into something called a locally compact abelian (lca) group. The Fourier transform, for instance, is really just a special map from the space L^1(S^1) of integrable functions on S^1 into the space c_0(\mathbb{Z}) of sequences which are indexed by the integers and which converge to 0 at both ends (these are none other than the Fourier coefficients of the L^1(S^1) function). It is possible to do Fourier analysis on other lca groups as well, and lots of nice things generalize (e.g. Poisson summation, the Plancherel formula, Fourier inversion, etc.). This is called commutative harmonic analysis. In fact we can also attempt to generalize Fourier analysis to nonabelian groups; this is the subject of noncommutative harmonic analysis. And harmonic analysis has profound connections to Lie theory, another beautiful area of math where groups are centre-stage.
And I could go on and on and on. Groups, rings, etc. are really every day objects for the mathematician, and this is why they're ubiquitous. They also form the foundation upon which the whole of mathematics rests.
Hopefully my ramblings here are useful (and also mistake- and typo-free!).
