While it was actually started by SNK, one of the high points of Capcom fighting games have been the character-specific intros that play up the characters' history and rivalries. With a whole new cast of fighters from Namco Bandai's Tekken joining the fray, Street Fighter X Tekken pairs off its fighters into sometimes-unlikely teams, with some oftentimes hilarious results.
Personal favorites of mine include Asuka and Lili pushing and shoving to figure out who's going to fight first, the total brofist moment between Ryu and Ken, and of course, THE SPACE CYCLONES. I'm a little disappointed that there aren't any special intros between Street Fighter and Tekken characters, though--at least not yet. What about you?