For example, is this language regular or context-free language?
On the one hand n can't be smaller than m but on the other hand you can't count n and m.
{(ab)^n (ab)^m | n>=m>=0}
That's a trick question, as that's just {(ab)^n | n>=0} written in a obscure way. Which is clearly regular.
In general, you prove that a language is or isn't of a certain type with a grammar, automaton, pumping lemma, the Myhill–Nerode theorem, etc.