Usenet is a collection of "newsgroups"...discussion forums, really, that dates back to 1979. Newsgroups can be read using a program called a "newsreader", or through a web site like Google Groups or Giganews. Many ISPs and some organizations still run "news servers", which receive messages from other servers.
Usenet is important in the history of the Internet because it's where a lot of the original "netiquette" rules were invented. It's also the source of a lot of historical Internet drama, like Cantor & Siegal (first mass commercial spam) and the Great Renaming.
These days, discussion has mostly moved onto the web. Usenet is mostly used as a way to distribute encoded binary files (**** & warez, mostly) away from the public eye and pretty much anonymously.