When you are in the atmosphere, you can turn your spacecraft (or airplane) by using the aerodynamic control surfaces (rudders, ailerons, etc.). These change the direction of the airflow around your craft. This makes your ship spin around, hopefully in a controlled fashion
While ascending, you'll notice that you'll start having trouble turning your ship. That's because there isn't enough air around your ship to throw around (in technical terms, the dynamic pressure is decreasing. The static pressure is also decreasing. The difference between these two is that static pressure measures the air... well, pressure, while the dynamic pressure measures the air flow, which increases with static pressure but also with speed). Eventually, aerodynamic control surfaces become pointless.
That's where the RCS (Reaction Control System) comes in. These are basically little rockets that turn your ship. For instance, suppose you want to pitch up (move your nose up). The front RCS thrusters will push your nose upwards and the rear thrusters will push your tail downwards. Of course, you can also pitch down, bank left and right and yaw left and right (there are quite a lot of RCS thrusters, both on the Space Shuttle and on the DeltaGlider, XR-2, etc.).
You can also use your RCS to move the ship without rotating it. This is called translation. Suppose that, instead of pushing the front upwards and the back downwards, the RCS thrusters push both ends of the craft upwards. End result is, you move upwards. Just remember that RCS thrusters are usually really weak, so only use them if you need a lot of precision, but not a lot of power (most obvious example, during docking).
Both of these two RCS modes are controlled via the numpad. The RCS is turned on/off by pressing Ctrl-/. By default, it's in rotation mode. You can switch between rotation/translation modes by pressing /. Refer to the manual for more detailed controls.