A preamp takes a low-level signal, such as a mic or guitar signal, and boosts it to the appropriate level (called "Line Level") for use in pro audio applications.

A power amp takes a line-level signal, amplifies it to an appropriate level, and converts it to the proper format for driving speakers.

Here is an example to illustrate the difference:

Guitar amps have two sections. The first section, where your guitar is plugged in, is the preamp section. This section takes the low-level signal from your guitar and boosts it to a line level signal. This section also typically includes the EQ, overdrive, and other effects. The "gain" control on your amp adjusts the volume of the preamp. After passing through the preamp section, the signal is ready for the next section, the power amp. The power amp section then amplifies the signal and sends it to the speaker(s).