It's short for "Tuscarawas County". A dark and boring county in Ohio located 2 hours directly south of Cleveland. Even the county's biggest towns (Dover and New Philadelphia) don't have much more to do than going to bars, eating food that's high in fat at restaurants, pacing around Wal-Mart and the mall, sitting at home playing video games, looking up random stuff on the internet, and sleeping. New Philadelphia has the Skate Place which is fun at first, then you realize it's full of middle school aged jocks. I know there's much worse places out there but T-Countiers try to make the county seem like more than it is, like they actually call Dover and New Philadelphia "cities". I wouldn't be too surprised if the county revolves it's entire economy around high school football, but I think it works as a good place to stop for gas, food, motel, etc. when on the way to another place. Dover just opened a new highway exit so that probably means more food places, gas stations, etc.
A family is taking Interstate 77
while on a road trip
Brother: Mom! I'm hungry!
Sister: So am I! I also have to go to the bathroom!
Mother: I just saw a sign saying were entering into Tuscarawas County!
Father: That's the perfect place to stop! I also need some gas!
...to sum it up, T-County is best as a place to stop!