That's exactly it! My designation here is "breeder" - generally, you're given either "trainer" or "breeder" as your job here. It should say on your trainer's card.

There are benefits to either class, but I rather like mine - I get to use HMs without a need for badges - although I guess that doesn't matter as much any longer since I have all the badges I'd need for them - and I tend to get more eggs when I breed my pokemon.

Basically you pair them up and encourage them to produce eggs! It's fun to figure out the best matches.