What are these 'chapters' of which you speak?
I don’t know all the details, because I haven’t actually read the books in which they’re really mentioned. But in some of the older Doctor Who novels (and I think possibly a few of the older televised stories as well), it is noted that the Time Lords were subdivided into Chapters (similar to the Houses in Hogwarts of Harry Potter). I believe there are six: Prydon, Patrex, Cerulean, Arcalian, Dromeian, and Scendles.
Apparently there really isn’t a ton actually said about them, but what little info there is, the fandom has taken and run as fast as they possibly can with.
If you look among the classic Who fandom blogs around here, a lot of things can be found under the tag #chapter blogging. Many of these things are amusing.
The main things you can find immediately are:
Prydonians are kind of the most talked about (the Gryffindors of Gallifrey, if you will). They have the most well-known Time Lords in their ranks, such as the Doctor, the Master, and almost every other renegade that’s worth mentioning.
Ceruleans frequently venture outside of the Capital dome. It’s generally assumed they touch plants, and are therefore considered fairly sketchy.
Patrexes wear heliotrope.
Arcalians are the science nerds.
I honestly don’t remember anything about the Dromeians, because nobody around here really talks about them. Their colors are kinda goth.
Scendles… well, we’re just broke. Champagne taste, dollar-store knockoff soda budget.