TITLE: Applications of class number
QUESTION [10 upvotes]: There is the notion of class number from algebraic number theory. Why is such a notion defined and what good comes out of it?
It is nice if it is $1$; we have unique factorization of all ideals; but otherwise?

REPLY [10 votes]: The class group of a number field $K$ can be used to parametrize other objects. 
1) If $[L:K] = n$, the possible $O_K$-module structure of $O_L$ is described by the ideal classes of $K$, although it is still an open question in general to show for each $n > 1$ and each ideal class of $K$ that there's an extension $L/K$ with degree $n$ such that $O_L$ as an $O_K$-module corr. to that ideal class.  (This is known for small $n$, but not for general $n$.) 
2) The orbits of the action of $\text{SL}_2(O_K)$ on ${\mathbf P}^1(K)$ are in bijection with ideals classes in $K$.  For instance, the action is transitive iff $K$ has class number 1. 
3) When $O$ is a quadratic order with discriminant $d$, the (narrow) class group of $O$ describes the primitive quadratic forms of discriminant $d$ up to proper equivalence.  Here we need a slightly more general concept than the usual ideal class group (unless $O = O_K$).
4) Weierstrass equations for an elliptic curve over $K$ up to a standard change of variables are related to ideal classes in $K$ (see Silverman's first book on ell. curves, Chap. VIII).