The style of writing a reply (via e-mail, newsgroup, forum, or other electronic communication method) with your response first, then the quoted material from the message you're responding to following below it, as opposed to bottom posting, where the quote comes first and are followed by your response.
The practice of top-posting is strongly opposed by most Usenet and Internet old-timers, because it puts question and answer in an illogical order (answer first, then question; some call top-posting "Jeopardy-style" because it reminds them of the popular TV game show), and because it's wasteful -- top-posting is usually accompanied by fullquoting, where the entire previous message is included without any snipping, including its signature block, and any ads or disclaimers that got tacked onto its bottom by servers it's passed through.
It gets even worse if a message has been through several rounds of top-posting and fullquoting, since not only the message being immediately replied to, but all earlier messages in the message thread's history, are included along with their associated signatures, ads, and disclaimers. Sometimes all of this material has been transformed back and forth between HTML and plain text, and lines split, joined, and re-word-wrapped in various ways, by various and sundry mail or news clients and servers, causing quite a mess. Eventually, the latest one-liner reply might have hundreds of kilobytes of trailing garbage.
If you read a mailing list in digest form, you've got to page or scroll through all the quoted material whether it's on the top or the bottom of a message, so people on such lists tend to demand the trimming of irrelevant stuff from a reply, something that just doesn't come naturally for top-posters.
Unfortunately, Microsoft Outlook, Outlook Express, and Hotmail (to name the Evil Empire's three major e-mail clients) do everything they can to encourage top-posting and put roadblocks in your way if you try to reply in any other style; for instance, signature blocks are always placed above the quotes, and if you're replying in HTML form (which these programs do when the message you're replying to is in that form even if you've otherwise configured it to use plain text only), it's nearly impossible to get the program to let you type below the quote without it being put in a really weird format that fails to distinguish where the quote ends and the reply begins. The obvious solution is to ditch those Microsoft programs, but it's distressingly difficult to get anybody to do this, even after they've had to recover from several Outlook-induced viruses.
In some corporate environments, top-posting is the ingrained "standard" and the PHBs are too clueless to understand a reply in any other format. Top-posting is actually well-suited to the types of correspondence that go on in a workplace; it lets the replyer almost completely ignore or misunderstand the specific points made in the message being replied to (where bottom-posting forces more point-by-point attention while you're trimming and responding to the quote), while covering his/her butt by quoting the whole thing back verbatim to show that he/she didn't actually ignore it or take it out of context (even though the reply makes it clear he/she really did). Or, less flippantly, it has value in such things as tech-support responses where a message thread regarding a particular problem might bounce back and forth between a customer and several different service reps; somebody getting a forwarded copy of the message after lots of this back-and-forth writing will have a whole history of the issue, though presented in the rather illogical reverse order (something like the movie Memento).
Another name for top-posting is TOFU, from a German phrase that's equivalent to the English "Text Over, Fullquote Under" (fortuitously preserving the acronym).