Anyone who knows anything about street gangs knows that it all comes down to turf. Gangs like to carve out their own little fiefdoms - identified in The Warriors through graffiti tags - and these represent home, a safe place to do business and a constant source of protection money from local merchants.
Above: Not exactly a fair fight, but that's what they get for invading
Of course, the way you lay claim to turf is by wresting it from other gangs, and rival boppers will try to muscle in on yours pretty regularly. If you want to hang onto what's yours while sending a message to anyone else who wants to try the same thing, you'll need to soldier up and kick the asses of whomever wanders in. You'll get your chance during the Turf Invasion missions, simple little defensive runs in which you'll have to beat down a whole crowd of punk-asses before the timer runs out. You won't actually lose your Coney turf if you fail, but you'll earn cool bonuses if you can protect it.