Tape trading is illegal, if by tape trading you mean making a copy of your copy and giving it to someone else in exchange for a copy of one of their tapes.
What you can do legally is make a personal copy of something for yourself. You can tape a show or movie off TV and in the U.S. you can even make a "safety" copy of a tape, DVD or CD you've purchased (in case the original is lost or damaged).
What you can't do is make copies of the copy, make multiple copies from the original source) or make your copy available for someone else to copy (like posting a file on your web site). Once you're doing that you're illegally distributing copyrighted material. This is the premise the recording industry is successfully suing illegal music downloaders under - even though they're not profiting from the illegal downloading it is still illegal activity.
It's important to remember that copyright law isn't just about profit too. One the basic ideas behind it is that a copyright owner should have
control over who sees their work, when, where and how. If they own the copyright and don't give you permission to copy/distribute their work then you don't have the right to. There's really no way around this except a little thing called "fair use."
The "fair use" clause in U.S. copyright law was created to allow things such as commentary, parody, journalism, research and education about copyrighted works without the permission of the author. Basically, they wanted to protect creators (people who own copyrights) but not infringe on freedom of expression.
Like I said before, if you're a journalist or do parody/critique as an artist (and generally you have to be accepted by the public as one of these - you can't just say you are an artist or reporter) you can make copies ONLY for the purpose of critizing the copyrighted work. You can also make use of short segments of a copyrighted work, but copyright should always be attributed.
For example, on
my site I posted a photo from CBC of their news anchor, Peter Mansbridge, recently. I did not get the CBC's permission to do this, but my use of that photo would generally be considered "fair use" because of three factors:
a) I deliberately chose to use a CBC publicity photo (not some candid shot or an unofficial one grabbed from Google's image search or another site)
b) I'm using the photo to help illustrate a discussion about a parody of Peter Mansbridge
c) I attributed the copyright on the to CBC.
If I was just using a PR photo on a site about Peter Mansbridge my use probably wouldn't be considered fair but because of point B (using it to explain or enhance a parody) I can get away with it.
Likewise, just attributing copyright on a site (point C) isn't enough. Slapping a "all material copyright Jim Henson Co." on a site won't make your use of Muppet photos, videos, etc. legal. If I had attributed copyright without point B I would probably be violating CBC's copyright on their photo.
It's also important that the majority of the content of my work (the web page) is not CBC's photo. If I had used multiple copyrighted photos of Peter Mansbridge that were similar, or posted a page of copyrighted Peter Mansbridge photos I would probably be exceeding the limits of fair use.
So you can see copyright is all very subjective. What your intent is and whether or not your activity is going to damage the commercial value of the work you are copying are factors that can only really be decided by a judge in a trial.
The problems in copyright law are defining what is and isn't fair use. Most of the activities people here claim they have loopholes for don't even come close to fair use. So if it's not "fair use" then you have to have the copyright owner's permission. Period.
I know most tape trading is harmless. Most people (here at least) seem to just want to get copies of stuff the JHC hasn't released on video or DVD. They're frustrated it's not legally available so the whole tape trading thing is understandable, but it's still considered stealing under the law.