Also known as a "username" or "handle", a Screen Name is simply a name associated with a specific member of an online community. The specific term "Screen Name" was invented by AOL as a more friendly alternative to "username," but today is widely understood around the Internet and is often used in forums.
Most Screen Names are pseudonyms, but there's usually nothing preventing a person from using his or her real name as a Screen Name if desired (unless it happens to be already taken). AOL's original innovation was that Screen Names could have spaces in them and so could in fact be real names, as opposed to the traditional alphanumeric (plus a few punctuation symbols, if you were lucky) traditionally used as account identifiers. At the time, a typical account ID on CompuServe looked like "72706,3526." On a BBS or UNIX system you might log in as "jsmith4."
Many, many Screen Names are copied from the user's favorite TV show, movie or song lyrics. This occasionally causes confusion when newcomers believe that a fan who took the name of their favorite actor or musician as a screen name actually is that person. If the Screen Name comes from the same source as that on which the online community is focused, it will usually have a number slapped on the end because numerous people want the name. Some people get incredibly creative with their screen names, creating phrases that would also double as names for rock bands.
Examples can be seen here.