Forgive the amateur question, I have read around and am still slightly confused.

A mixer is something were you connect the instruments pre-sound card.

An interface doesn't even need to be connected to a sound card.
a mixer is something that mixes the sounds of multiple channels together. it can be either a physical component or a piece of software. it lets you set the levels of each input, as well as some basic (or not so basic on high end models) EQing and panning. the idea is to take multiple inputs and send them out as a single mono or stereo channel. this can be done for recording or live sound.

an audio interface is something that gets your signal to the computer and back out. your sound card is an audio interface. the ones usually mentioned for recording on UG are USB or firewire, but PCI or PCIE cards are still audio interfaces. the idea here is to get your sound from an external source onto the computer to be manipulated, and in the case of recording, preserve the highest quality you can in the process.