I have three (somewhat related) examples:
1) The notion of explicit construction. Seeking explicit constructions to non-constructive replace existence proofs is an old endeavor. Computational complexity offers, in some cases, formal definitions (constructions that can be dome in P or in polylog space.) But these definitions are slightly controversial. In any case people looked for explicit constructions before any explicit definition for the term explicit construction was known.
2) The notion of effective bounds/proofs. There are many important problems about replacing a proof giving non effective bounds with a proof giving effective bounds. Usually I can understand a specific such problem but the general notion of effectiveness is not clear to me. (A famous example: effective proofs for Thue Siegel-Roth theorem.)
3) Elementary proofs. I remember that finding elementary proofs for the prime number theorem was a major goal. I was told what this means many times and in a few of those I even understood. But the notion of "elementary" proof in analytic number theory remained quite vague for me.
