A mixer lets you combine multiple sources (microphones, lineboxes, guitars, recorders, computers etc.) to one (or more) outputs, like a main PA system, a recording interface, a monitor mixer etc.) Mixers come in sizes ranging from as little as two inputs to hundreds, depending on its application, and can have many different effects built in.
A microphone converts changes in air pressure (experienced by humans as sound) into electrical impulses, that can then be recorded, played back and manipulated. Microphones come in many shapes and sizes to fit different applications. The most common types used in studio and live recording of music and spoken word include dynamic, condenser and ribbon. Some require external power, also known as 48v or Phantom power (like condenser) while some should under no circumstance be phantom powered (ribbon), so be careful before you engage the 48v button on your mixer or interface.