Let's say you want to record a full band; drums, bass, guitarist and vocals. You first have the mics, various kinds for various jobs. These mics will usually use XLR to XLR cables to connect them to the mixing desk's mic preamps.
The desk has 2 main volume controls, the channel gain knob at the top and then the channel fader at the bottom. In between are various EQ, FX and AUX (auxiliary send) channels.
Usually when recording you aim to get the best sound possible using correct mic placement. Sometimes you may have to tweak the EQ a little or add a small amount of gentle compression - but once these tweaks are recorded they can't be undone without the band replaying the whole song again, so it's usual to add effects AFTER to the playback from tape or your PC.
The desk needs to have multiple channel outputs that you then connect to your multichannel soundcard via the apropriate cables - this varies from model to model. Some newer desks use a multichannel digital output direct to your PC via USB2 or Firewire - such as the new Mackie Onyx range.
On the PC you need some kind of multichannel audio sequencer package to record and then edit the song. Examples are: Cubase, Sonar, Pro Tools or Logic (for Macs). Using this software you'll record the seperate instruments to individual channels. You can then manipulate the tracks using EQ and other effects and dynamics processors.
Ultimately when you're happy with how the song sounds you mix it down to a final stereo file, typically a high quality (24 bit) wav file. This is then mastered to tweak it and make it sound perfect.
MM