Well there are a lot of different things, but there are a few obvious ones. Try to bring a friend who knows something about cars too. Even if they don't know anything, another set of eyes is always good.
Check under the car right away for puddles of any kind of liquid. Common but sometimes serious leaks are oil (dark brown), transmission (usually red), or coolant (all kinds of colors, mostly blue or green). Stay away from cars with big puddles of any of these.
When you start the car, pay attention to how the car turns over, or starts. It should be smooth and responsive, and you can easily tell if it is not. Ignition and electrical problems can be a real pain.
Also, press down on the gas for a couple of seconds, then get out and check the exhaust. Considerable amounts of oil coming out the exhaust is a huge red flag. Shift the car through all of its gears, and make sure the transition is smooth and natural.
When you're actually driving the car, keep in mind that all cars feel a little different, but some things should never happen. When you have a safe area to drive, usually on a dealer's empty lot, accelerate to like 15 mph then let go of the steering wheel for a few seconds. If the car consistently turns itself one direction or the other, it could be a sign of a serious alignment problem.
Acceleration and braking are pretty unique for each car, so its sometimes hard to nail down symptoms of problems from just one test drive. Try to feel how smooth the car accelerates, especially when changing gears. Brakes should never feel too unresponsive, and if you have to slam on them they are probably too worn out.
Trust your instincts, you can usually tell a good car from a bad one. Don't be afraid to walk away from any used car, there are plenty more out there.