Pre-amps are devices that supply power to microphones and boost the signal received from microphones.
A pre-amp can be used as a stand-alone power and output source for a microphone when little other equipment is available.
Mixers combine several audio signals into a single audio output.
A mixer can be used as a pre-amp as well as a hub for multiple microphones.
Both mixers and pre-amps can be used as stand-alone equipment, but using both can greatly increase the audio. A mixer should be used if multiple microphones are being utilized, however.