a mixer controls the sound input/output. You can also change the EQ settings if possible. Picture:
http://www.dv247.com/invt/547
MIDI interface is a unit that controls your midi devices. You plug the midi interface directly into your computer and then plug your midi controllers into the interface via MIDI cables.
Picture:
http://www.dv247.com/invt/9264
Audio Inteface is basically your soundcard. These can come in either PCI, USB or Firewire, depending on your choice.
You dont need all 3 if you want to start recording. If you want to record audio tracks, i would say get a souncard. A midi interface is essential if your soundcard doesnt have MIDI connections on.
Hope that helps a little.