The melo- part of melodrama comes from the Greek melos, which also gives us the word melody, and a melodrama was originally a stage play that had an orchestral accompaniment and was interspersed with songs. Historically, melodramas dealt with romantic or sensational topics.
These days, though, the melody has gone from the melodrama. Modern melodramas don't necessarily have any music; they deal with stereotypical characters and exaggerated conflicts and emotions. You've probably seen people in plays or on TV shows who swoon (faint) when they hear bad news — that's a mainstay of melodramas. (But have you ever seen anyone faint from bad news in real life?)
Melodrama refers to exaggerated emotions off the stage, too. When someone overreacts to a situation ("I said 'hi' to Bobby and he didn't say 'hi' back! I'm a social outcast!"), they're being melodramatic.