"Territory" is a plain, factual word. It's the one you should use if you are not sure. It is good standard English. It has a very wide range of application. It can refer to a national or regional boundary. It can refer informally to an area of ground a person or group treats as belonging to them. It even has uses by biologists in animal behavior--some bird songs, for example, are actually warnings that say "this is my territory."
"Turf" literally means a piece of firm, grassy earth. "The neighbor with the new house is using fertilizer on his lawn in order to establish the turf."
There are several other meanings by extension.
"Turf" meaning "territory" is an informal word that has entered the language from slang used by street gangs. it is used specifically with respect to bureaucratic infighting within large organizations. "People don't really use Powerpoint to communicate information, they use it as a weapon for fighting turf wars." "Don't fight with Sandra about needing to get authorization from her department, she's just defending her turf."
"Turf" also informally refers to.... horse-racing!
And, finally, one very specialized, yet surprisingly common use in U.S. restaurants: "Surf and turf" is a popular menu item that combines a small lobster (surf) with a piece of steak (turf)! Personally, I think it's a weird combination, but in restaurants with "plain American cooking" lobster and steak are typically the two most expensive entrees, and "surf and turf" is an obvious choice for someone who wants the "best" but can't make up their mind.