How to prove that the complement of a regular language is always regular?
If a language is indeed regular that means there is an FA that accepts it. The complement of L is just the language of all words that are not in L. 

Thanks to Rick Decker for mentioning in the comments that this only works for FAs that are deterministic and to D.W for correcting the answer. 

Now, a trick we can perform to test that the complement of L, namely L', is actually regular is to take the FA that accepts L and reverse all final states to non-final states and all non-final states to final states. Note that start states in the old FA become start and final states in the new FA. 

This new FA will then accept all words present in L' which are words not in L.

In conclusion, take the FA accepting L and then form a new FA by:


Changing all final states to non-final states 
Changing all non-final states to final states 


The new FA accepts all words not in L, which is the language L'.