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.