In algebraic number theory, the nicest rings are perhaps ones with unique factorization. For Dedekind domains, this is equivalent to being a principal ideal domain. In this talk, we introduce a tool for measuring "how close" certain Dedekind domains (specifically rings of integers in a number field) are from being a PID; namely, we define the ideal class group and discuss a proof that it is finite. We will also do some nice and not-so-nice examples.