How to prove that a language is not regular? We learned about the class of regular languages $\mathrm{REG}$. It is characterised by any one concept among regular expressions, finite automata and left-linear grammars, so it is easy to show that a given language is regular.

How do I show the opposite, though? My TA has been adamant that in order to do so, we would have to show for all regular expressions (or for all finite automata, or for all left-linear grammars) that they can not describe the language at hand. This seems like a big task!

I have read about some pumping lemma but it looks really complicated.

This is intended to be a reference question collecting usual proof methods and application examples. See here for the same question on context-free languages.
Proof by contradiction 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:


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


To prove that a language $L$ is not regular using closure properties, the technique is to combine $L$ with regular languages by operations that preserve regularity in order to obtain a language known to be not regular, e.g., the archetypical language $I= \{ a^n b^n \mid n \in \mathbb{N} \}$.
For instance, let $L= \{a^p b^q \mid p \neq q  \}$. Assume $L$ is regular, as regular languages are closed under 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 $, $I/a^p= \{ a^{r}b^rb^p\mid r \in \mathbb{N} \}=I.\{b^p\}$. All classes 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.