How you get sound out of your mixer and connect it to your audio interface depends on how elaborate your mixer is.

Using a Basic Mixer

If you have a basic mixer that only features stereo outs, the mixer outputs to separate left and right channels. You will only be able to patch those two channels into your audio interface. Using the pan controls on the input channels, you can send certain sounds to the right channel and certain sounds to the left channel.

Using a mixer like this doesn't mean you lose total control of the sounds. By panning certain sounds all the way left or right, you force that signal into one of the two cables. Each of the output cables can go to a separate track on your recorder. Your mixer may have the ability to work as a USB or FireWire interface. With one of these mixers you will generally only be able to send the main stereo output channels from your mixer to your computer, and receive a stereo signal back to your mixer from your computer. The same panning ideas from above apply to these mixers.

More Sophisticated Mixers

If you have a mixer with multiple output buses, you can use those to connect your mixer to your audio interface. This gives you a greater degree of flexibility in how you can set up your recordings. For every bus you have, there will be another two outputs to your recorder. A decent-quality four-bus mixer will give you a total of four extra channels of output to your recorder. If you pan correctly and set the bus assignments well, you can route tons of signals flexibly to your recorder. If you use a lot of microphones and your recorder doesn't support many microphone inputs, this might be a cost-effective way to go—it's cheaper than buying all those microphone preamps separately.

Let's say you're recording drums, bass, guitar, keyboard, and vocal. Using a mixer that only offers you a stereo out, you must either record the instruments separately (or two at a time with the guitar and bass) or get the perfect balance of all the parts going into your computer. Now let's say you have a mixer with four stereo busses (or eight total outs) and an audio interface with eight inputs. With this setup you could devote three outs to the drums, one to the guitar, one to the bass, two to the keyboard, and one to the vocal.