What would starting a project be like, and what are some examples of projects?

Starting a project requires some serious time and/or dedication, and checking here might be a good use of your time; although it's a bit old, breeding and projects hasn't changed all that much since then. Basically, you have a set goal in mind (max stats, a special design, some combination of the two, etc etc), and you breed a series of Sylesties together in order to get that goal.
Some examples of projects would be Articu's various projects, Scath's long-running Sunbather Gazania max-stat project, LadyMoriarty's Hallowed Moon Kelparies, and you can probably find more while browsing around the Breeding Discussion forum or in people's stables. I have an on-hold breeding project in the second tab of my stable.