This is a big list of applications of the class number formula and its generalizations. I'll start:

 1. The solution to [Gauss's class number problem][1] for imaginary quadratic fields, and more generally the [Brauer-Siegel theorem][2].
 2. The comparison of the class number of a field with the class numbers of its subfields.
 3. The application of the Beilinson-Bloch conjecture to the arithmetic Bogolomov-Miyoaka-Yau inequality. See [here][3].
 4. The explicit construction of class fields of totally real fields via [Stark's conjectures][4]. 

What else?


  [1]: https://en.wikipedia.org/wiki/Class_number_problem
  [2]: https://en.wikipedia.org/wiki/Brauer%E2%80%93Siegel_theorem
  [3]: https://terrytao.wordpress.com/2007/05/04/distinguished-lecture-series-iii-shou-wu-zhang-%E2%80%9Ctriple-l-series-and-effective-mordell-conjecture%E2%80%9D/
  [4]: https://en.wikipedia.org/wiki/Stark_conjectures