Audio mixers can go from the sheer simplicity of a mono summing mixer, with a bunch of inputs and a single output with a knob to control output volume to the complexity of a multi-bus digital mixer with more built-in DSP power than a small recording studio. So, it's kind of a big topic, and we'll try and break down a few basic categories to get you started.
Line mixers are designed to handle, as you might suspect, line level signals. Most frequently, they're used as sub-mixers for things like multiple keyboard setups on stage or simple summing mixers. The things to look for in a line mixer are flexibility in connections and quality internal electronics—high-quality, low-noise op amps and capacitors. They tend to be fairly stripped down—no EQ, no inserts, no DSP—but are compact and very simple to use.
Larger mixers, in addition to more channels, will start to have features like multiple sends, subgroups, and built-in DSP for effects. The EQ will also get more sophisticated and you'll begin to have assignable routing on channels and returns.
Digital mixers start to hit the crossover point between live and studio mixing, with some intended to do both. These mixers have analog to digital conversion at the inputs and do everything in the digital world once the signal is in the board. They will have a USB or FireWire output so you can record the stereo mix directly to a computer system. They'll have more internal processing power, with some even doing some internal guitar amp or microphone modeling, giving you more options for sound. A convenient feature of many of these boards for live sound use is the ability to store presets, so you can easily recall the settings for a particular venue, saving a lot of setup and sound check time when you come back to someplace you play regularly.
Many new mixers have connections to tablet computers, so you can use a tablet for remote mixing and touchpad control.