tv - To whom did Jaime Lannister lose on a previous tournament that people talk about in season one and what was the tournament honoring?

I'm going to hazard an educated guess and say that you're talking about the tourney that Jaime lost to Loras Tyrell. This was only of note because Petyr's dagger (implicated in the attempted murder of Bran and assault of Catelyn) was allegedly lost to Tyrion because of the upset.

I don't believe we were given many more details on the tourney than that, i.e. who it was in honour of.