In its simplest incarnation, online publishing is the publication of material in an electronic format. This encompasses several forms: magazines ("zines" or "ezines"), novels, scholarly journals, Usenet (newsgroup) postings, and Frequently Asked Questions files (FAQs), to name a few.
The most important thing to keep in mind is that the state of online publishing remains in constant flux. The rules are literally being made up as we go along, especially in the areas of copyright, ownership, and distribution. If you're a writer, a good general rule is to assume that by posting your story, poem, essay, or whatever to any publicly-read area (such as a newsgroup or web page), you're "publishing" it, and therefore may not be able to offer certain rights (such as First North American, for example) if you try to sell it to a print magazine. More detailed information can be found at "The Internet Writer Resources" Page, or in the misc.writing FAQ.
A common misconception--and major hurdle to the idea of online publishing as a legitimate venue--is that publishing a piece online is somehow "lower" on the totem pole; that if you publish your story online, it means you can't hack it in the real world. After all, if you have a computer, modem, and Internet access, it's extraordinarily easy and inexpensive to make your story available to thousands of people. This is the idea behind rec.arts.prose, a newsgroup repository of uploaded writing that, in many cases, hasn't been edited or even proofread. (I won't get too deep into this, because, honestly, I haven't spent much time exploring what's available there. As with most everything related to the Net, I'm sure there are gems in that mineshaft, but you probably have to do quite a bit of digging to find them.) With the growth of online zines and work being done by the people involved with Project Gutenburg (volunteers who are creating libraries of electronically stored books that can be downloaded for free), however, online publishing is gaining credibility.
And momentum. Once word got out that thousands of people are reading online materials, many of the major print publications began scrambling to establish a presence on the Net. Although this bandwagon mentality has resulted in many hastily- and poorly-executed digital shadows of otherwise respectable print publications, it has also created a movement towards forming standards by which works can be protected, and authors paid for their efforts. These standards are still only flickers on the digital horizon, but unless the Net collapses or the Earth burns up, they will have to be defined to accommodate this movement.
So, should you cancel your subscription to the New Yorker or Time?
Has the day finally arrived when bookstores will fold, and we'll do all of our
reading on laptops and Newtons? Of course not. I think the people who have been
predicting the end of the printed word are sitting too close to their monitors
without proper shielding. People aren't going to want to buy a half-dozen
batteries along with their copy of The Newly Repainted Bridges of Madison
County or Newt's current bodice-ripper. Besides, you just can't put a cup of
coffee on a well-worn electronic reading tablet without worrying about the
warranty. 
For a more in-depth discussion of online publishing, see L. Detweiler's excellent "A Vision of the Future." Or, feel free to wander the many links to writing- and literature-related sources.
