Fun Bobby was Monica's boyfriend on and off in Season 1 and Season 2. He first appears in "The One With The Monkey", when he shows up as Monica's date to her New Years Party. He does not live up to his nickname, however, when he shows up in a flood of tears and reveals that his grandfather has just died and, not being able to get a flight out, decided to show up at the party.
In Season 2, Monica gets back together with Bobby, who now has problems with alcohol. Monica successfully helps him give up drinking, but once he does, he is not fun in any way. Chandler nicknames him "Ridiculously dull Bobby" and Monica begins to drink heavily around him so she can be somewhat entertained by him. Bobby tells her that he thinks she has a drinking problem and breaks it off because he is "not capable of being in a co-dependent relationship right now".
Bobby's drinking problem was most likely a result of his grandfather's death, suffering from depression without alcohol as a result, which would explain why he became so serious.