Newsgroups date back to the very earliest days of the internet. They were developed to enable people at different universities to talk to one another about subjects of mutual interest. A (very) brief history can be found here.
There are several newsgroups which are relevant to mathematicians; the most important and general are sci.math and sci.math.research.
The newsgroup sci.math is available at Google Groups and at Mathforum. The moderated newsgroup sci.math.research is also available at both locations.
The FAQ for sci.math can be found here (another version lives here).
The moderated group, sci.math.research, has information available here.
These groups are, in theory, a wonderful way to exchange information and give and receive help from other mathematicians and interested amateurs around the world. However, the fact that sci.math is not moderated - in other words, anyone can post to it - can mean that useful information is deluged in a sea of junk. People post their supposed elementary "solutions" to Fermat's Last Theorem, there is a never-ending battle to convince doubters that Cantor's diagonal argument really does work and that the real numbers really are uncountable, and some stuff which doesn't look like mathematics at all appears there.
The other newsgroup, sci.math.research, is moderated, which means that posts must be approved before they appear. This means that there is much less mindless dribble there, but it also makes for a less active newsgroup. The activity that does take place there is more focussed, however.
Having said all of this, there are some genuinely good and wise mathematicians who frequent the newsgroups. These people often have the answers.
To increase your chances of getting these answers, it often helps to: 1. state the problem clearly. 2. check to make sure that the question you're asking isn't a Frequently Asked Question and 3. make it look like your question isn't an attempt to get strangers to do your math homework (the newsgroups don't like to do your homework and resent being asked to do it).
I hope that any newsgroup experiences you have are good.
