I have written out units, counits and their triangle identities for the few common examples of adjoint functors (tensor-hom, free-forgetful...) but that didn't remove my confusion completely.
I've tried to think about my confusion and I think it all boils down to these questions:
Where does the name unit/counit come from? Is there a connection with units of algebras/counits of coalgebras?
I always see examples of adjunctions between functors written out using hom-sets (and units/counits are there, lurking in the background). Is there an example where adjunction is written more naturally in the unit/counit language than in the hom-set language?
Continuing from the question before, it seems I just don't have a good idea of their use. Are there proofs that are easier to do by using units/counits?