Proof by contrapositive is often used to show that a language is not regular: let $P$ a property true for all regular languages, if your specific language does not verify $P$, then it's not regular.
The following properties can be used:

 1. The pumping lemma, as exemplified in [Dave's answer][1];
 2. [Closure properties][Closure] of regular languages (set operations, concatenation, Kleene star, mirror, homomorphisms);
 3. A regular language has a finite number of prefix equivalence class, [Myhill–Nerode theorem][MyhillNerode].

To prove that a language $L$ is not regular using closure property, the technique is to combine $L$ with regular language in order to obtain a languages known to be not regular, e.g., the archetypical language $I= \{ a^n b^n | n \in \mathbb{N} \}$.
For instance, let $L= \{a^p b^q | p \neq q  \}$. Assume $L$ is regular, regular languages are closed by complementation so is $L$'s complement $L^c$. Now take the intersection of $L^c$ and $a^\star b^\star$ which is regular, we obtain $I$ which is not regular.

The Myhill–Nerode theorem can be used to prove that $I$ is not regular.
For $p \geq 0 $, $I/a^p= \{ a^{r}b^rb^p| r \in \mathbb{N} \}=I.\{b^p\}$. All class are different and there is a countable infinity of such classes. As a regular language must have a finite number of classes $I$ is not regular.

 
[Closure]: http://en.wikipedia.org/wiki/Regular_language#Closure_properties
[MyhillNerode]: http://en.wikipedia.org/wiki/Myhill%E2%80%93Nerode_theorem


  [1]: http://cs.stackexchange.com/a/1032/12