TITLE: What are cohomology rings good for?
QUESTION [18 upvotes]: I am studying some concepts of algebraic topology myself, and I read lately a bit about cohomology rings (created by the direct sum of cohomology groups) but besides all definitions I could not find any information.

What are they good for? 
How do we use them? 
Could someone give me a very short descriptive example?

Thank you

REPLY [12 votes]: One reason that the cohomology (and homology, and homotopy group) functors are so important is that they give a means of testing how similar two spaces are. For instance, the $\pi_1$ functor suffices to distinguish $\mathbb{R}^1$ from $\mathbb{R}^2$, up to homeomorphism (by removing a point); more generally, the $k$th homotopy (or homology) group will distinguish $\mathbb{R}^k$ from $\mathbb{R}^m$, $m > k$. 
Now cohomology, homology, and homotopy groups (and K-theory and other cohomology theories) are all functors from some category of spaces to some algebraic category. Presumably, the idea is that the algebraic category will be easier to think about -- one can compute explicitly, at least if the functor applied to nice spaces produces simple results (which often happens).
At the same time, if these are to be a powerful tool at distinguishing spaces, we want these functors to take values not in the simplest category (e.g. the category of sets) but a category with more structure: groups, or, better, rings. The cup-product shows that the cohomology functor takes its value in the category of rings.
This means in practice that if you have two spaces with similar looking cohomologies as abelian groups, you may still be able to tell that they are different if the ring structures are different. As an example, you can see that complex projective space $\mathbb{CP}^2$ and $S^2 \vee S^4$ are not homotopy-equivalent, even though their homology and cohomology groups are all the same. The reason is that the cup-product is not trivial in the first, while it is trivial in the second.
So the more algebraic structure you can put on a functor from spaces to sets, the finer this functor will be as a tool to distinguish spaces. Cohomology has this nice ring structure, but in fact it has more: it turns out that the cohomology ring (with $\mathbb{Z}/2$-coefficients) is a module over the Steenrod algebra.