Races are exactly what they sound like; pick a "track" somewhere within Liberty City and see who can be the fastest. Those races, however, are for LAME people. If you want to do something that's actually interesting, GTA Races are centered less around winning, and more around making sure everyone else loses. You can pick from an assortment of vehicle types, but you're not confined to them, and you're free to leave your car, grab a different car or grab a bus and completely block the track. GTA Races feature weapon pickups, too, so get ready to pop some windows and explode some rivals.
Turf War is essentially a control-point match; as a member of one of two gangs, it's your job to capture "turf" - represented by a glowing marker that shows up on your map - by standing on it until control of it switches to your team. It's fun, so long as you've got at least a few people who don't mind hanging back for a while and actually defending what you've captured.
This one's sort of like Mafiya Work, only instead of random objectives, you're tasked with tracking down specific cars, stealing them from their defenders and driving them back to a rendezvous point. The green ones are full of drugs, and therefore worth more. Think of it like Capture the Flag, except that you're not stealing from the other team directly.