Basically if you are using an IC and you leave a pin floating, it's value can oscillate wildly and cause problems in your circuit.

A weak pull up is a situation where if the pin is left floating, it will be pulled to logic high (it follows that a weak pull down would pull the pin to logic low). If an external signal is applied to the pin, it only takes a small amount of current to pull the pin low, and as such it behaves pretty much like a normal pin when a signal is applied.

Weak pull ups and downs are usually configured in software and are particularly useful when attaching switches to micro-controllers as a pull-up/down resistor is no longer required.