A public address amp is designed for making announcements over small speakers with limited range. It may be designed for a 70 volt system that has transformers at the speakers so that the wire may be small gauge and be connected in parallel without damaging the amp. It will typically be single channel although some are designed for background music as well and these might be stereo. Power will vary with the number of speakers.
A power mixer is the combination of an input mixer and power amplifier in one package. This allows for easier mobility. The amplifier part will vary with the application. As with the pa amp functional with minimum quality needed.
A home theater or stereo amplifier will be 2 to 7 channels with any power rating you need or are willing to pay for. 50 watts per channel to 1000 watts per channel. Quality of sound is usually key to high end audio amps and many home theater amps.