There are two basic types of online "text" discussion, asynchronous or "on-going" and synchronous or "real-time" discussion (also known as chat). In asynchronous on-going discussion, there is a delay between when one participant sends their discussion text and when the other participant(s) receives that text. In synchronous or "real-time" discussion, the discussion text is sent immediately so that the recipient sees it soon after it was typed. For more information on this type of discussion, see: Real-time Discussion.
There are a number of technologies for having asynchronous on-going discussion. Email was one of the first ways to do this. Online forums is another popular form of on-going discussion. Finally blogs provide a way for on-going discussion through blog comments.