Originally it referred to a chain of postings on a single subject in a newsgroup. Most newsreaders include a command that lets you follow a thread by jumping to the next related message (rather than reviewing all the messages in order). Popular newsreaders also have a thread selector that allows you to sort articles by threads; indentation is often used to indicate a response to an article positioned above it.
In addition to threads occurring on discussion boards, forums, and BBS', threads also appear on blogs. Most recently, however, this term has morphed into a common expression to describe a series of e-mail messages. When you "reply" or "forward" an e-mail to someone else and include the original message below a new one, it creates a series of messages which become a thread. As with the newsgroups, the prior messages are often indented or tagged with brackets. This usage is commonly seen in work-related e-mail because certain parties "cc" each other and will later forward the thread to someone not on the original distribution list; the top message may say "Be sure to follow this thread to get the background info on it."