To begin simply, tape trading is a method of sharing music among music fans. The music being spread is most often what the general public thinks of as bootlegs, i.e. recordings made by fans at shows. However I make a distinction between bootlegs and concert tapes. I use the term bootlegs to describe recordings that are being sold without the knowledge of the artist. Whereas concert tapes are traded freely among fans. You can read my opinions on the distinctions between bootlegging and tape trading elsewhere.
As I said before, tape trading is largely based on recordings that have been made by fans at concerts. It all starts with those tireless tapers, who use all their knowledge and tricks to get copies of these shows. The taper usually uses a small tape recorder, either digital or analog, and a tiny microphone, similar to the ones news anchors wear. This is necessary because most venues do not allow taping, and therefore the taper has to hide the equipment on the way in, and keep it inconspicuous during the concert. Depending on the equipment being used, the position of the taper, the quality of the sound system, and whether or not she's standing next to some drunk guys, the taper can come away with a very good quality recording of the concert.
Even better than this audience recording is a soundboard recording. This is when instead of using a microphone, the taper somehow or other manages to plug the recorder directly into the venue's soundboard. Obviously this takes the permission of the sound person. Either that or you have to be the sound person. These types of tapes are rarer, but are much better quality, because they consist more of the music, and less of the guy next to you yelling "Play some Jerry, man!" In essence, that is how concert tapes get made. They can also be demos or unreleased tapes leaked from the studio, or rare promotional items. In essence, anything that you can't go out and buy is a possible object of trading.
I never encountered tape trading outside of the internet, so I don't know if it occurs very widely there. And I've only been following trading for a few years, so I don't know it's distant roots. But I think it's safe to say that the Deadheads are the core of the tape trading movement. Not only in numbers and longevity, but in their enthusiasm and spirit. Tape traders can sometimes be ugly, materialistic and crass, but the Deadheads seem to live up to the ideals of generosity and sharing. They and the rest of the tape traders are a group of people who are willing to trade copies of their concert tapes in order to get new concert tapes. It's a win-win situation. Especially when you consider the cost of a blank tape versus the cost of a bootleg CD. And especially when you consider that outside of the tape trading world, this music just isn't available. Artists release one or two albums a year, yet they perform scores of concerts in that same time, some much more daring and exhilirating than their released material. Concert tapes are preserving some music history that would otherwise be forever lost.
Tape traders communicate through newsgroups, the web, and mailing lists. Find a site, group or list focused on your favorite artist and you will probably find people trading concert tapes. Of course if you are a beginning trader, you don't have anything to trade, am I right? But with a little perseverance, and depending on what musical circles you inhabit, you should be able to find someone willing to start you off. When I first joined the Neil Young mailing list I found a host of people willing to make me tapes. All I did is send the blank tapes to them, and thank them when I got the concerts back. This is ideally how things should work for newbies. Unfortunately there are a lot of people out there who will only make tapes in return for something else. In other words if you don't have tapes to trade they will demand that you send them two tapes for every one that they send back, or that you send extra money. Many people do just that, but if I can put my two cents in (and who's going to stop me?) I would recommend avoiding any such offers. I have never had to pay or send extra blanks for any of my tapes and I've done just fine. Keep looking and I'm sure you will find a generous benefactor somewhere (heck, it could even be me).
I'll leave you with that much information about tape trading. There's much more to know about it of course. For some specifics, you might want to read my guidelines. If you have any additional questions, feel free to e-mail me.