Really, it depends on the setting. For example, what makes for an awesome boss fight in an RPG wouldn't work so well in an FPS. However, one thing that stays constant is a feeling of "holy cow, this is going to be awesome!". Some examples, showing how it differs for me between genres.
1. Solomon Grundy in Batman: Arkham City- This boss fight was awesome to me, because here I am, as the goddamn Batman[sup]tm[/sup], and I'm fighting, on even ground, with an 11 foot tall undead monstrosity. The feeling of dread, combined with excitement, really made it stand out.
2. First Ridley Prime battle in Metroid Prime 3: Corruption- This one stood out to me because of the sheer scale of it. There you are, as bounty hunter Samus Aran, in a powersuit, but still human (well, somewhat), fighting against a dragon-like enemy, completely in free-fall. That was just an adrenaline rush.
3. Dhoulmagus in Dragon Quest VIII- This one is a bit different. It wasn't the scale of the battle, but rather, a feeling that all the battles, all the fighting, everything, had led up to this moment, and feeling like the fate of the world hinged on my winning this fight, against an opponent that was not going to go down easy, and was in fact tougher than any boss up to that point.
1. Solomon Grundy in Batman: Arkham City- This boss fight was awesome to me, because here I am, as the goddamn Batman[sup]tm[/sup], and I'm fighting, on even ground, with an 11 foot tall undead monstrosity. The feeling of dread, combined with excitement, really made it stand out.
2. First Ridley Prime battle in Metroid Prime 3: Corruption- This one stood out to me because of the sheer scale of it. There you are, as bounty hunter Samus Aran, in a powersuit, but still human (well, somewhat), fighting against a dragon-like enemy, completely in free-fall. That was just an adrenaline rush.
3. Dhoulmagus in Dragon Quest VIII- This one is a bit different. It wasn't the scale of the battle, but rather, a feeling that all the battles, all the fighting, everything, had led up to this moment, and feeling like the fate of the world hinged on my winning this fight, against an opponent that was not going to go down easy, and was in fact tougher than any boss up to that point.
He wasn't the final boss, but to a first time player, it really felt like he was.