Here are my top 10 favorite boss fights:

1. Alma's Awakened Form (Ninja Gaiden)
2. Tower Knight (Demon's Souls)
3. Mizar (Jet Force Gemini)
4. The End (Metal Gear Solid: Subsistence)
5. Phalanx/The Thirteenth Colossi (Shadow of the Colossus)
6. Ornstein and Smough (Dark Souls)
7. Kefka (Final Fantasy VI)
8. Jack Krauser (Resident Evil 4)
9. Kazuya Mishima (Tekken 2)
10.  Vergil (Devil May Cry 3)

• Red Falcon (Contra)
• Ugh-Zan III (Serious Sam: The First Encounter)
• Vile (Mega Man X)
• M. Bison (Street Fighter Alpha 3)
• Jubileus (Bayonetta)
• Spider-Man and Batman (Revenge of the Shinobi)

My favorite boss battles:
1.Sephiroth
2.Bowser (Paper Mario-N64)
3.Dagoth Ur

Thardus in Metroid Prime is pretty awesome. You have to use the Thermal Visor to find his weak points. The whole Metroid series has some pretty awesome bosses in general.

Yakuza 4's final boss. Every final boss ends on some sort of rooftop in the Yakuza series. On Yakuza 4, you have four characters, and between the dialogue and bosses themselves, the battles in Yakuza 4 are my favorite.

Honorable mentions.

Lords of Shadow 2 Victor Belmont, Aria of Sorrow Julius Belmont, and Symphony of the Night Richter Belmont. The battles aren't very difficult, but the Belmont battles are my favorite in the Castlevania franchise. Richter Belmont on Portait of Ruin isn't included because the way the AI is designed in that battle, really doesn't feel that exciting of a battle.

Every other last boss in the Yakuza series. The rest of the final bosses in Yakuza are almost as good as that of Yakuza 4. Yakuza 3's last boss is better than every last boss on Yakuza 4, but the combined four on Yakuza 4 is what gives them the edge over Yakuza 3's last boss.

Every boss from Shadow of the Colossus,

Vergil from DMC3 and DmC Devil May Cry.

YES!

That battle was amazing.  When Bowzer rose up in the distance I had no idea what to do at first.  He was just so big!

Desert dragon, chasing that with Agro, making the jump and climbing onto it's back, doesn't get more epic than that!

Medusa - Kid Icarus Uprising

All other boss battles I played were not epic or beyond. Dat music during her battle made it epic alone....then the whole battle itself made it awesome.

edit:

Final Boss in DQ 9, and when I had to fight a Dragon just after my ally dragon was killed. Sad, epic moments.

WHole game was awesome and I want the new ones to come to the west.

Stallord, thank you very much!

Best boss ever indeed, though Nintendo is actually very good and original at bosses in general. The Zelda series alone contains many classics for me (like OoT Ganondorf for sure). I also like Bowser in both Yoshi's Island and Mario 64 a lot.

Further on I'd like to add;

Desann in Star Wars: Jedi Knight II

Probably the most epic Lightsaber battle in game history. The perfect finale to a great game, with a perfect setting, a perfect musical score and a prefect intro only surpassed by the monologue given by Darth Malak in Knights of the Old Republic. Desann himself, due to the nature of the game, is a better battle though.

SA-X in Metroid Fusion

This character had been the player's nemesis for the entire game, and it did a very good job of being fearsome. At the end, Samus is finally powerful enough to battle it, but it is still very hard. The fast paced battle itself and the twist of the ending make this a very memorable boss and defeating it is very satisfying.

Zarok in MediEvil

Last but not least, this boss is not among my 'favorites' because it was such a good fight, in fact, it's an awkward and unbalanced one, but rather because it was probably the most ridiculous and funny end-boss I ever witnessed. Talk about memorable. The whole game is very lighthearted, but the designers went truly went all out with this boss.

The first Titan in Unreal for the first time in 1998.

Jet force Gemini boss on n64 had me pulling my hair out. when I beat him the sense of accomplishment was amazing.

gannandorf in Ocarina of time. That battle was intense and sad at the same time. you realized this amazing game was coming to its close.

And lastly I had great memories from beating super metroid. That brain was insane.

honourable mention to gears of war 3. I just loved that game, but the end battle was sad and memorable.