We seem to be a little short on the 'and why' part of the thread title, here…
Luffy
"You guys go. He's mine"
Of all the great battles Luffy has had, the one I remember most is his fight with Ao Kiji. A challenge so great, even Luffy couldn't cope, and he knew it, sending his friends away to safety before risking his life. It was sobering to see his Storm move, once so magnificent, doing absolutely nothing to this powerful enemy.
Zoro
"There are blades in this world that can cut anything at all…"
Zoro vs Mr. 1. Possibly my favourite battle in the series, despite the fact that the rest of Zoro's battles tend to be rather forgettable. From the moment he rushes in (to save Nami, no less- a rare moment of unabashed heroism from Zoro) to discovering the 'Breath Of All Things', the fight had me gripped. I wince every time I see Zoro getting drilled, and he gets hit again while he's lying helpless…truly magnificent, epic, and filled with suspense.
Also, he throws a house, which is on JUST the right side of the awesome/ridiculous scale.
Nami:
"Compared to what Vivi's feeling…this pain is nothing!"
Not a lot of choice here, so Miss Doublefinger. Better in the manga, but with a lot of good moments (The shock before the mirages are revealed, getting spiked through the foot, the judo move she does with her cloak) Somewhat spoiled by slow pace and unnecessary jiggling in the anime, though.
Usopp
"I'm going to beat you, Luffy…I know exactly how I'm going to do it!"
Heartbreaking as it was to watch the fight between two friends, Usopp vs Luffy was the first time that Usopp impressed me with his skill, determination and ingenuity. Clearly, he works a little better with forward planning.
Sanji
"..That traitor Nami's next…you understand me, hero? You aren't saving anyone.""
The fight at Arlong Park against Kuroobi, without a doubt.
Sanji's pain, claustrophobia and panic underwater is one of the most emotionally effecting scenes in the series as far as I'm concerned.
The release of tension, as Sanji, on dry land, takes down his enemy in ten action-packed seconds, is made all the more glorious by what has come before.
Chopper
"If you cry, you can't call yourself a man!"
I've always liked Chopper's fights, mostly because, unlike most of the show's characters, he really doesn't like fighting very much. When you put an innocent character like him in a combat scenario, you see how horrible it really is. You can see this clearly in his fight against Kumadori at Enies Lobby…I feel so sorry for the little guy, he's trying so hard to protect his friends, and he finds he just isn't strong enough. Of course, then Monster Chopper comes along and I start feeling sorry for Kumadori, but that part wasn't so good.
Robin
"Know the weight of the history you have destroyed!"
Although there is obviously a limited selection here, Robin vs Yama has always been a particular favourite of mine. There's something very striking and memeorable about the image of this thin, elegant figure pounding the hell out of this gigantic, bloated monstrosity. Also, this is the first time we see her lose her temper and use her powers to the fullest, and it's an impressive sight.
Now that I think about it, her comments about never being able to replace the past gain extra meaning when we know about the tragic events of her earlier life. I wonder if that was intentional?
Franky
"You call that a punch?"
While Franky has yet to especially impress me in battle, the boxing match with Fukurou is always good for a laugh. It's manly and bloodly stupid at the same time.