I too am a DJ and had a difficult time understanding all the tech stuff. I used to use a Tascam Porta 7. I was looking into getting a digital 8-track, but I realized that my money was better spent on getting a computer. I got together w/some knowledgeable buddies and basically built the computer from scratch (most of the name brand computers don't give as much bang for your buck as computers that you build yourself).
If you do use your computer, you'll basically insert the computer into your setup pipeline. Instead of outputting your DJ mixer directly to your amp/monitor, you output into your computer's soundcard. You can then output from the sound card to your amp/monitor. I know that for most of you out there this is really, really basic, but it seems like this is the starting point that he/she was looking for. You'll need some software to record your audio (I use N Track). The software will give you more than 8 tracks (assuming your computer can handle the load).
Again, this is basic for most of you, but I know what it's like to be completely lost. Lyrical, do you understand the dynamic of tracks? On a tape-based multitrack, there are 4 tracks. There are two tracks on each side of the physical tape. When you record on the 4 track, you are essentially working with two tracks on each side of the tape, for a total of 4. Let's say you record on track 1, you are recording on 1/2 of the tape on side A. Recording track 2 is equivalent to recording on the other 1/2 of side A. Track 2 is essentially and physically beside track 1 on the tape. When you play it back, the multitracker simply plays both tracks at the same time. Add track 3 and track 4, and the multitracker will play all four tracks. Does this make any sense? On a computer you can have several tracks. For example, on track 1 record a beat. On track 2 record a scratch over that beat. On track 3 record some movie dialogue. You can play back all 3 tracks at the same time. You can add more tracks over those. It's just adding layers. I hope this helps.
By the way, your question re 4 track v. 8 track brought the Mixwell "Fundamentally Sound" mixtape contest to mind. Is that what sparked your interest?