The fact that a commutative ring has a natural topological space associated with it is still a really interesting coincidence.  The entire subject of Algebraic geometry is based on this simple fact.

Question: Are there other categories of algebraic objects that have interesting natural topologies that carry algebraic data like the Zariski topology on a ring (spectrum)?  If they exist, what are they and how are they used?  

Yes, there are plenty of such things.  

[In the following, "compact" implies "locally compact" implies "Hausdorff".]

1) To a Boolean algebra, one associates its Stone space, a compact totally disconnected space.

(Via the correspondence between Boolean algebras and Boolean rings, this is a special case of the Zariski topology -- but with a distinctive flavor -- that predates it.)

2) To a non-unital Boolean ring one associates its Stone space, a locally compact totally 
disconnected space.

3) To a commutative C*-algebra with unit, one associates its Gelfand spectrum, a compact  space.

4) To a commutative C*-algebra without unit, one associates its Gelfand spectrum, a locally compact space.

6) To a commutative Banach ring [or a scheme over a non-Archimedean field, or...] one 
associates its Berkovich spectrum (the bounded multiplicative seminorms).  

7) To a commutative ring R, one associates its real spectrum (prime ideals, plus orderings on the residue domain.)  

8) To a field extension K/k, one associates its Zariski Riemann surface (equivalence classes of valuations on K which are trivial on k).  

This is by no means a complete list...

Addendum: I hadn't addressed the second part of your question, i.e., explaining what these things are used for.  Briefly, the analogy to the Zariski spectrum of a commutative ring is tight enough to give the correct impression of the usefulness of these other spectra/spaces: they are topological spaces associated (cofunctorially) to the algebraic (or algebraic-geometric, topological algebraic, etc.) objects in question.  They carry enough information to be useful in the study of the algebraic objects themselves (sometimes, e.g. in the case of Stone and Gelfand spaces, they give complete information, i.e., an anti-equivalence of categories, but not always).  In some further cases, one can get the anti-equivalence by adding further structure in a very familiar way: one can attach  structure sheaves to these guys and thus get a class of "model spaces" for a certain species of locally ringed spaces -- e.g., Berkovich spectra glue together to give Berkovich analytic spaces.   
