I loved it, it's a while since I've seen it, but a basic summary is that the strange gypsy voodoo woman used to be with Davy Jones, and she was a sea god. She put him in charge of ferrying people who die at sea to the other side... but something happened (I forget what, I think it was that she got imprisoned as a human form, maybe?), so he gave up on what he was meant to be doing, and started collecting people on his boat, and he cut out his heart and put it in a box.
Anyway, the posh guy from the East India Trading Company wants no more pirates, so since he has Davy Jones' heart (end of POTC2), he persuades Davy Jones to destroy all the Pirate Ships with his big monster-under-the-sea...
The main cast, and the woman who is actually Calypso (sea god) go to Davy Jones' locker (afterlife) and save Jack because all the pirates are being killed and some condemed pirates called for a brethren court, but Jack never named a sucessor to him, so they have to bring him back.
Yeah, the court decides to remove calypso from her imprisoned form. Causing a big battle around a whirlpool, between some people from East India Trading, and the black pearl, and Davy Jones' ship.
By this point, the heart is with one of the parties in the battle, so that becomes something that they're all fighting over. Will wants it so that he can 'kill' Davy Jones and become captain of his ship (which must always have a captain) 'cause his dad is part of the imprisoned crew, and Jack wants the immortality it brings him. East India Trading wants it cause it gives them power over Davy Jones.
Anyway, during the battle, Will gets stabbed and Jack allows him to have the heart so that he can stab it and become the captain of Davy Jones' ship and ultimately live forever.
The End.
I'm sure some people will be able to correct bits of that.
Admittedly, it is confusing. I think you have to have seen the second one not too long before to really follow it, the two seem to go together in one story.
From what I've heard, the fourth one is the start of a second trilogy. (They're going back to basics with it, new story and the such. I think because of the way the 3rd ended and because of the amount of cast that dropped out, they kinda have to start something new.)