I don't know about you, but bosses are a big deal for me in games. Awesome boss fights can make a game for me even if the other parts are ho-hum, and if the boss fights aren't to my tastes, the game most likely won't become a favorite of mine.
What do I look for in a boss fight? (of course, feel free to completely disagree)
1. Something different. A boss shouldn't just be a normal enemy with some more health and stronger attacks. The fight should shake things up, push you out of your comfort zone and force you to think in a different way. This is why bosses in first-person shooters are usually nothing special. Like in Borderlands - the bosses looked good and interesting, but overall they were glorified normal enemies, usually surrounded by actual normal enemies. Another serious offender is Assassin's Creed II and its spinoffs.
2. Reasonable size. I really hate when the bosses are too big. Some (many) games think that they need to make the bosses as big as possible for them to be impressive or challenging. But for me, no - a towering giant of a boss doesn't feel personal. These fights are usually limited to one of two paths: either attack the legs/some other weak point in the body until he falls and then attack with everything, or climb the boss and look for weakpoints without falling off. Some offenders are Bayonetta (which was a nice game but had lame bosses), Shadow of the Colossus (obviously. I found most of the fights ultra boring) and God of War (some of the bosses are awesome, but the most seemingly "impressive" fights didn't hit the spot).
3. Some kind of connection to the characters or story automatically makes the fight better. Of course, you can't expect all the bosses to be like this (I'm okay with the generic "huge scorpion" or "badass guardian of the divine kingdom" bosses here and there), but the dramatic, game-changing battles shouldn't be against someone you haven't seen (or at least heard of) before. An even bigger sin is breaking this law in the final battle of the game. I'm looking at you, Final Fantasy IX and Mass Effect 2.
I've named some games with disappointed bosses, but which ones do I like?
Many bosses in The Legend of Zelda. Some examples are Dark Link (Zelda II and OoT), Ganon (LttP), Goht (Majora's Mask), Twinrova (OoT).
Many bosses in the Metal Gear Solid series. To name a few: Sniper Wolf, The End, Psycho Mantis, The Fury. The idea of bosses corresponding to one unit (like in MGS and MGS3) is great for point 3, and makes the game that much more memorable. MGS4 really disappointed me with its boss fights.
Many bosses in the Souls series. Penetrator, Flamelurker and Capra Demon are great examples. Also Priscilla and Nito (I'm a sucker for grim reapers).
Speaking of grim reapers,I love how you get to fight Death in every Castlevania game, even if the fights aren't always that exciting.
Bosses in Chrono Cross. Especially Dario and Miguel.
Most of the bosses in Rayman Origins.
They aren't exactly bosses, but I liked fighting the Dragon Priests in Skyrim. They almost made me forgive the awful fighting mechanics for a while.
There are many more, but I want to hear what you have to say, especially if you have some bosses which fit my criteria.