I don't really understand what people mean with "breeding projects". Are they trying to make the pet look a certain way?

Yes!

A breeding project involves working to breed a certain pet, with certain traits/colors. There are several ways to do this. The first (and most common) way is to use essences, which will allow you to design a pet of the species of the essence, to look any way you want. You can also choose the traits of a generated pet, up to the gp limit on the essence (2gp for regular, 4gp for enhanced and mystical.)

I.e I have a breeding project in the works right now, with this as my end goal:



To get there, I've generated two pets with the traits I want, in the colors I want:



and I'll breed them together until I get the 4carry pet I need to advance in my project. Xav has a phenomenal guide on the mechanics of it here.

The other way is to breed pets of any color/lineage to get a certain set of traits, and then to recolor the end result. I don't have any personal experience in doing this, so I can't provide any images here, sorry;;

Does this answer your question? It ended up being a lot longer than I anticipated orz

Yes, thanks a lot.

One of my successful breeding projects, the 'Midnight Ride Kelpari"




^this is Calypso



^and this is Sol.

and breeding projects are fun :3