The feeling you get when someone keeps doing the thing you have asked them not to do, and your patience finally runs out. It is a mix of anger and tiredness, with a little helplessness in it — you have tried, and it has not worked. It is an uncountable noun, so you say a lot of exasperation, not "an exasperation". You will meet it most in a few fixed shapes: in exasperation ("She threw the instructions down in exasperation"), with exasperation ("He sighed with exasperation"), and a note / sigh / look of exasperation. It is a formal word — nobody says "I felt exasperation" out loud; they say "I'd just had enough". Notice too that it is often shown, not stated: writers use it as the label for a sigh, a raised hand, a look at the ceiling.