If you are looking for a brand new car, I was impressed with the redesigned 2006 Hyundai Accent. Although the price creeps up into the $13,000ish range, if you wait until the year end clearance and bargain hard, you could probably drive away from the dealership for around that price. The Accent has a long warranty, pretty good interior feel from my experience, and doesn't seem so bad. The redesigned Kia Rio is basically the same car.
I would personally skip the Chevrolet Aveo because for the same amount you could have an Accent or Rio which seem to be worlds ahead of the Aveo.
I have seen deals on brand new Suzuki Forenzas for around $11,000, but then again, for my money, I'd rather have the Accent.
If you are looking to buy used, I would look for a clean, off rental fleet, 2004 or 2005 Ford Taurus, Chevrolet Malibu, or Chevrolet Classic. Those cars are dowdy, average, but inexpensive to buy and inexpensive to repair should you have to eventually.
If you don't mind the higher mileage, you could get a 2000-2002 Saab 9-3 SE hatchback in that range. It is a sorta presitigious car, will drive quite nicely, is fairly stylish, but the reliability on any Saab is iffy, and maintenance costs after 100,000 miles can be astronomical.