As far as I am aware, they are used to ensure a node is at a particular level rather than just flapping around without a connection.
The circuit on the left demonstrates a pull down resistor (R1). It is used to ensure the base of the transistor is at 0v when the switch is open, otherwise the base would not be connected to anything and a that is supposed to make a transistor's behavior unpredictable, so when the switch is pressed, the base is supplied with current and the lamp lights up. In the circuit on the right, R3 is a pull-up resistor, it is used to pull the base of the transistor up so that current flows into the base when the switch is open, turning the transistor on and lighting the lamp, when the switch is closed, the current instead flows to ground and the base falls to 0V and the lamp turns off, the opposite effect the the circuit on the left.
More experienced members please feel free to correct me, as I'm only a novice myself.
**broken link removed**