At the end of Episode III: Revenge of the Sith Padmé died in childbirth. According to the medical droid Padmé was (Medically at least) perfectly healthy so according to most people Padmé died from a broken heart... or did she?
Keep in mind this is just theory. Their are bonds between living beings known as force bonds, these are stronger between force sensitive beings such as Jedi. It is possible that Anakin Skywalker and Padmé Amidala had a force bond, either unwittingly created by Anakin through his love for Padmé or purposely created by Darth Sidious so he could use Padmé to control or influence Anakin.
At the end of Revenge of the Sith there is a duel between Obi Wan Kenobi and the recently fallen Anakin a.k.a. Darth Vader. At the end of this duel Anakin was left close to death whilst Padmé was perfectly health but by the end of the movie Anakin/Vader is as close to healed as is possible for him whilst Padmé dies. Vader asks about his wife but Palpatine reveals that she is dead despite Vader still feeling her through the force.
My theory is that (Possibly unwittingly) Darth Vader used a combination of his force bond to Padmé and a dark-side force ability similar to Force-Drain to drain Padmé's life force to replenish his almost entirely drained life force, so effectively it's Pamé's life force flowing through Vader's body. This would explain: how Darth Vader survived after losing both legs and being burned alive; how Padmé died despite being perfectly healthy and how Darth Vader could sense Padmé after her death.