On the Japanese side of the spectrum:
Samurai Champloo - The series finale. Mugen vs. that lunatic with the sickle on a chain inside the old abandoned church. There are moments during that fight that any aspiring animator should study; it's phenomenal. And from the same episode - I consider them one fight because of the way the episode is edited - Jin vs. Kagetoki Kariya, especially the part when they're fighting on the pier. It's so fast and so precise - look for the moment when Kariya's blade comes so close to Jin's face in one near miss that it takes his glasses off.
Street Fighter II: The Animated Movie - Vega vs. Chun Li in her apartment. She gets out of the shower and sits on the bed to braid her hair, Vega literally rains from the ceiling and the two of them tear that apartment to shreds for the next three and a half minutes. Also, compare the original Japanese version of the fight to the American dubbed version. Each is uniquely relevant, in that the American version is pulse-pounding and empowering on the part of Chun Li, while the Japanese version is essentially you watching Vega rape her. It's the exact same scene, but the music completely alters your mood as the viewer. It's pretty intense and very scary.
End of Evangelion - Asuka in Unit 02 vs. the Eva Series. I can't say enough great things about this fight. The drama is palpable, the danger is everpresent, the tension is nail-baiting as her Eva is running out of battery power during the entire brawl, and the animation is some of the best I've ever seen. The attention to weight and scale is breathtaking.
On the American side:
The Great Mouse Detective - Basil vs. Ratigan inside (and out) of the famous Clock Tower in London. This one is the gold standard for me. No American animated feature to date has managed to quite pull off just how plain violent this fight is. I saw this fight as a child and was blown away by how visceral and aggressive it was, especially for a Disney film. I wasn't cheering; I just wanted it to be over. I wanted somebody to save him. Nothing else (animated in America, anyway) has ever made me feel that way. The staging is spectacular and the whole scene just oozes drama. It's got something to do with the fact that, at this point, Ratigan's master plan has failed, and all that matters now is killing Basil. I love that kind of conviction. It's not about good and evil in this scene; it's just personal. Very personal. It remains my favorite fight scene to this day, even among the anime I mentioned.