A mixer is needed where (1) your player or amplifier does not have a microphone input; or (2) when you want to mix sound from multiple sources; or (3) when you want to add features to your sound system, such as key control, vocal cancelling, echo. The mixer is placed between your player and your amplifier.
A powered mixer is a mixer that includes/integrates an amplifier and possibly speakers. A powered mixer is quick to set up, and so is good for smaller groups and venues. This option is also called a "mixing amplifier."
A digital mixer can store and recall your setup, or a template for setting up for a particular type of performance. A non-digital mixer may be easier to set up in live situations.