OPENING:

Final Fantasy VIII gets this. Hand downs the best introduction to any game I've ever played ever...EVER.

In a closer second is Tekken 5. The awesome intro where Kazuya and Heihachi take on an army of Jacks, ending with Kazuya betraying Heihachi by throwing him into a pile of Jacks. Kazuya subsequently is able to escape while his father is unable to get free. The island blows up as the Jacks set themselves to self-destruct. Awwwwweeessssoommmme.

ENDING:

Because every character in a fighting game has a different ending, all my best endings are from that genre of gaming. There's just a lot more to love.

Tekken 3 had a funny-ass ending for Ling Xiaoyu. Heihachi promises Ling he'll open a theme park in her name if she wins the tournament. Obviously she's done so because you just finished the game with her. Heihachi flicks the switch, however, and the park name is revealed to be "Heihachi Land". Ling then proceeds to pwn Heihachi.
JERK!

SoulCalibur 3 had a pretty good ending for Cassandra. I'm biased toward her obviously but it's still pretty funny. Cassandra returns from her quest to destroy Soul Edge having paid a severe price. She had broken the sword she had "borrowed" from the shrine of Hephaestus, the god of forge and fire. She tries to sneak the sword back into place but her sister catches her and makes her fix it. Cassandra cries because she can't.