Some of my favorite Boss Battles in games and what I think makes them great

– Luca Blight (Suikoden 2)

A great example on how to translate a huge threat in the story to make sense in terms of gameplay. Blight is one of the most evil and despicable characters ever, and he's also very powerful. His cruelty, ruthlessness and prowess in battle is established and built up very well throughout the game until you finally fight him for real, after managing to lure him into a trap.

And the fight does not disappoint. He's easily the most dangerous enemy you will face for a long time. It takes three whole parties (18 people!) to finally bring this monster down.
At the end, he's cornered and shot with multiple arrows, but even then – as a wounded, bloody mess – he's still not defeated. It takes one final duel with the hero to kill him for good.

What makes this fight great is that it's not just another boss battle, which it easily could have been. This whole fight is a big difficulty spike and it feels like a real event compared to most other, run of the mill RPG boss battles. They did a great job of doing this villain justice and accurately portraying him as the threat that he is in gameplay as well as story.

– The End (Snake Eater)

The End is easily one of the most well-known bosses in the Metal Gear series.
He kind of betrays the standard of what we expect from a "boss fight", and it's pulled off pretty well. I think that's what made him memorable for many.

The fight takes place in a huge arena (Even separated through loading screens) and features no music, you just hear the sounds of the jungle as well as other sound effects. You're required to actively seek your enemy out, which can be done in a variety of ways.

It's a tense encounter that's quiet and calculated, focusing more on stealth and observational skills. Although it can be cheesed in a lot of ways too, overall there is just a lot to this battle and its rich design along with a TON of little details makes it one of the highlights of the game.
I think its main advantage is that it brings something different to the table and kind of turns the traditional boss battle on its head by featuring slow and calculated gameplay, which suits the game mechanics better anyway.

– Knight Artorias (Dark Souls)

This is a boss that's more satisfying from a mechanical standpoint. It doesn't try to reinvent the wheel or feature any special gimmicks, it's just a straightforward fight.

That doesn't mean it isn't brilliant, however. Artorias really captures the fun of encountering a challenging opponent, learning their moveset, getting better and better at fighting them and then finally rising to the challenge and beating them.

Apart from the satisfying moment-to-moment gameplay, I think the struggle is what makes fights like this fun. A demanding, but fair fight like this can do wonders to invest you in the game.
After digging deep into the mechanics of the fight and putting in the work you're rewarded with a great sense of accomplishment that I'm sure many fans of the Souls series in particular know very well. From Soft is really good at designing these kinds of fights, and there's many others I could have listed here (Ornstein & Smough, Slave Knight Gael, Ludwig the Accursed, Guardian Ape, …).

I think Artorias in particular has a very unique, but intuitive moveset along with striking the perfect balance of challenging, but fun.

– What are some of your favorite bosses? And what do you think is it that elevates them above so many others?