Couldn't Vader save his son and continue being a Sith Lord? Couldn't he have let Luke go and basically become the Emperor and have the Galaxy to himself? I don't think killing the Emperor to save your son will automatically convert you to the light side. Enlighten me.
The point of being on the Dark side is that you're motivated by "dark" emotions: Anger. Possessiveness. Powerlust.
Saving Luke was the direct opposite.
Vader is in a constant struggle between his Dark Side tendencies and the remnants of Anakin pulling towards the Light Side - we see this in the new Disney canon book "The Sith Lords", where
Vader saves a little girl from being killed by Darth Sidious. (mind you, in the end Palpatine wins and that little gesture leads to her entire village being killed - by Vader, on his orders).
The short answer is: Because he sacrificed himself out of love for his son, who had just spared his life when he was defeated and helpless.
For the full answer, we need to begin with a little background.
When Luke is preparing to surrender to Vader, he stops to tell Leia the good news ("I'm your brother!") and the bad news ("Vader is our dad!"):
Luke: Then you know why I have to face him.
Leia: No! Luke, run away, far away! If he can feel your presence, then leave this place! I wish I could go with you.
Luke: No, you don't. You've always been strong.
Leia: But why must you confront him?
Luke: Because there is good in him. I've felt it. He won't turn me over to the Emperor. I can save him. I can turn him back to the good side. I have to try. [kisses Leia on the cheek, then leaves]
Luke finds an Imperial soldier and surrenders; he is taken to Vader, but the family reunion doesn't go quite the way Luke had planned:
Darth Vader: The Emperor has been expecting you.
Luke Skywalker: I know, father.
Vader: So, you have accepted the truth.
Luke: I've accepted the truth that you were once Anakin Skywalker, my father.
Vader: [angry retort] That name no longer has any meaning for me.
Luke: It is the name of your true self. You've only forgotten. I know there is good in you. The Emperor hasn't driven it from you fully.
[steps off to the side of the passageway, away from Vader]
That was why you couldn't destroy me. That's why you won't bring me to your Emperor now.
Vader: [inspects Luke's lightsaber] I see you have constructed a new lightsaber. Your skills are complete. Indeed you are powerful, as the Emperor has foreseen.
Luke: Come with me.
Vader: Obi-Wan once thought as you do. You don't know the power of the dark side! I must obey my master.1
Luke: Your thoughts betray you, Father. I feel the good in you, the conflict.
Vader: There is no conflict.
Luke: You couldn't bring yourself to kill me before and I don't believe you'll destroy me now.
Vader: You underestimate the power of the Dark Side. If you will not fight, then you will meet your destiny.
Luke: I will not turn...and you'll be forced to kill me.
Vader: If that is your destiny.
Luke: Search your feelings, father. You can't do this. I feel the conflict within you. Let go of your hate!
Vader: It is too late for me, son.2 The Emperor will show you the true nature of the Force. He is your master now.
Luke: Then my father is truly dead.
There are two crucial lines here, but they are very easy to overlook:
1 Vader's choice of words here is important. He says "You don't know the power of the Dark Side. I must obey my master!" Why is this important? Because it means that Vader doesn't obey Palpatine because he likes him, or because he wants to do it. He obeys because the Dark Side is too powerful to resist. He is trapped.
2 Again, when Luke says "Let go of your hate!", Vader doesn't say "No way! Hate is the best thing ever, and I love it!" He says, rather mournfully, "It is too late for me, son. The Emperor will show you the true nature of the Force. He is your master now." This gives me the impression that:
Vader regrets becoming a Sith. He doesn't like being Vader. He feels that his bad decisions in the past have doomed him to remain evil, and if he had the opportunity, he would go back and do things differently.
If Vader could leave the Emperor's service, he would do so, but he thinks he isn't strong enough to pull it off, or good enough to deserve a better life. He hates himself.
Luke is brought before the Emperor, and Vader provokes him into losing his temper and attacking Vader in blind rage. After a brief but vicious battle, he regains his composure after lopping off Vader's hand.
He realizes that killing Vader would be tantamount to becoming Vader. More importantly, he has finally realized what no one else knows: Vader isn't some all-powerful, monstrous evil overlord. Behind the scary black mask and the robotic armor, Vader is really just a fragile, cowed old man who has no control over his own life. Taking pity on his wounded and humiliated father, Luke tosses his light saber aside.
Luke proudly declares his loyalty to the Light Side of the Force, and throws the Emperor's derision back in his face. The Emperor is enraged, and unleashes a barrage of Force-Lightning. This is the moment in which Luke's plan is revealed: he is utterly defenseless against the Emperor's attack, or at least appears to be, and he begs his father to help him. He has just shown unexpected mercy on his father, and that act is what sets the wheels in motion.
Vader is torn between his fealty to the Emperor and his newfound love for his son - his son who just moments earlier chose to let him live, when he was incapable of defending himself and it would have been incredibly easy to kill him. The sight of his own, unexpectedly merciful son writhing in agony, mere moments away from death, is too much for Vader to bear. He makes the ultimate sacrifice. He lifts his master above his head, carries him to the edge of the abyss, and hurls him in. In the process, he is mortally wounded by the now-undirected Force-Lightning.
Let's look at how the script describes this scene:
Luke: Never. I'll never turn to the Dark Side. You have failed, Your Highness. I am a Jedi, like my father before me.1
Palpatine: [angrily] So be it... Jedi!
Luke refuses to give in to Palpatine's demands and disarms himself.]
Palpatine: [raising his hands toward Luke] If you will not be turned, you will be destroyed! [Fires lightning bolts at Luke, causing him to fall to the floor in agony. Darth Vader gets up and stands next to Palpatine, watching.] Young fool. Only now, at the end, do you understand. [shoots another round of lightning] Your feeble skills are no match for the power of the Dark Side! You have paid the price for your lack of vision! [continues shooting lightning]
Luke: [writhing in agony] Father, please! Help me!
Palpatine: Now, young Skywalker... you will die. [intensifies lightning blasts]
Darth Vader/Anakin Skywalker2: [looks back and forth at Luke and Palpatine. He grabs Palpatine from behind and throws him down into the reactor shaft]
[A redeemed Anakin Skywalker is dying in Luke's arms]
Anakin Skywalker: Luke, help me take this mask off.
Luke: But you'll die!
Anakin: Nothing can stop that now. Just for once, let me look on you with my own eyes.
[Luke carefully removes Vader's mask to reveal Anakin's disfigured face underneath.]
Anakin: Now...go, my son. Leave me.
Luke: No, you're coming with me. I won't leave you here. I've got to save you!
Anakin: [smiles] You already have, Luke. *You were right. You were right about me...*Tell your sister... you were right...
[The redeemed Jedi, Anakin Skywalker, dies]
More important details which are easily overlooked:
1 Luke's choice of words is very clever, and serves two purposes:
He compares himself to his father, but surprisingly, he does so in a favorable way. Despite everything Vader has done to blacken his own name, Luke is proud to be his son. Vader surely must have taken note of this and began to realize the incredible truth: In spite of the fact that Vader led Luke to the Emperor, expecting that Luke would either be corrupted by the Dark Side or killed outright, Luke loves him.
It also reminds everyone - especially Vader - that Luke's father is a Jedi. Vader hasn't thought of himself as a Jedi for 20 years. This probably made him think about his life before and after he fell under the evil influence of Palpatine. For the first time in decades, he remembers that life wasn't an endless nightmare before he fell to the Dark Side.
2 The script, which suddenly switches here from calling the character "Vader" to calling him "Anakin", makes it clear that this is the exact moment when Vader ceases to exist and Anakin is reborn. Update: I haven't been able to find the script I quoted here - most versions of the script I can find keep calling him "Vader". It is possible that the script I used was altered by a fan.
One of the novels gives us some insight into what Vader was thinking while the Emperor was ravaging Luke with Force Lightning. Vader's thoughts are in italics for the sake of clarity:
Vader watched Luke curl into a fetal position as the Emperor hurled an even more staggering wave of lightning at his victim. Vader had no doubt that Luke was about to die. His son screamed.
Not just my son…
The Emperor unleashed another round of lightning.
…or Padmé's son…
Luke screamed louder.
…but my son… who loves me.
Luke's clothes began to smoulder as his body involuntarily spasmed. Suddenly, Vader realised he was no longer concerned about his own personal future. Despite all the terrible, unspeakable things he'd done in his life, he knew he could not stand by and allow the Emperor to kill Luke. And in that moment of awareness, he was Darth Vader no more.
He was Anakin Skywalker.
It took all of his remaining strength to seize the Emperor from behind. The wretched Emperor continued to release lightning bolts, but they veered away from Luke and arced back to crash down upon him and his insurgent apprentice.
- The Rise and Fall of Darth Vader
Vader was redeemed by his son's love, which inspired his own love for Luke. He sacrificed his life out of the love of Luke. His motives, as much as his actions, were the essential factors that allowed him to redeem himself. He had returned to the Light Side of the Force by forfeiting his life.
With that out of the way, we can finally see what George Lucas has said about the subject:
"It really has to do with learning," Lucas says, "Children teach you compassion. They teach you to love unconditionally... He doesn't right the wrongs, but he stops the horror. The end of the Saga is simply Anakin saying, I care about this person, regardless of what it means to me. I will throw away everything that I have... and throw away my life, to save this person. And I'm doing it because he has faith in me; he loves me despite all the horrible things I've done... he still cares about me, and I can't let that die. Anakin is very different in the end... He takes the one ounce of good still left in him and destroys the Emperor out of compassion for his son."
- George Lucas, The Making Of Revenge Of The Sith; page 221
"You learn that Darth Vader isn’t this monster. He’s a pathetic individual who made a pact with the Devil and lost. And he’s trapped. He’s a sad, pathetic character, not a big evil monster. I mean, he’s a monster in that he’s turned to the Dark Side and he’s serving a bad master and he’s into power and he’s lost a lot of his humanity. In that way, he’s a monster, but beneath that, as Luke says in Return of the Jedi, early on, “I know there’s still good in you, I can sense it.” Only through the love of his children and the compassion of his children, who believe in him, even though he’s a monster, does he redeem himself."
- George Lucas, quoted in J. Windolf, “Star Wars: The Last Battle,” Vanity Fair, 2005
"And obviously there are two sides to the redeemer motif in the Star Wars films. Ultimately Vader is redeemed by his children.
- George Lucas
Why do people think that Darth Vader turned to the light at all? My opinion:
- Killing your master is totally a Dark side thing (in EU now I can't think of any Sith not to trying to kill his master or a more powerful Sith).
- Trying to save a beloved person (his mom and later Padme) is what brought Anakin to the Dark Side at first. Saving the person you love or your family is not a accepted thing for a Light side, it is totally connected with the Dark side.
I think Vader hadn't been redeemed, he just chose to kill his master for his son, and that, according to Jedi code, is totally a Dark Side choice.
Just something to add here; there seems to be a bit of contention over whether or not killing the Emperor was a light side or dark side move, as killing one's master is generally a no-go.
"A Jedi uses the Force for knowledge and defence. Never for attack." This quote is from Yoda, and I think it plays an important part in how the scene between the Emperor, Vader, and Luke plays out.
In the scene, the only one who uses the Force is the Emperor. He uses it extensively as an offence, only for attack, when electrocuting Luke.
Vader is known to use the Force for such things, especially force-choking and throwing people and objects around.
Luke doesn't so much use the Force in this scene, and in fact abandons his lightsaber once he realises his emotions had taken hold and realising that it jeopardises him, proving himself as a true Jedi, though somewhat weak in the control over his emotions which may compromise him.
Darth Vader, one who relies often on his use of the Force against people, goes in hands-first to pick up and throw the emperor. I think this change from his habitual Force use, which would mark him as a lord of the dark side, to using only his own strength, is what may attribute to his transition back to the light side, and indeed back to Anakin.
This, coupled with the arguments above, is why I agree with the idea that Vader ceased to exist in this moment, and he was once more Anakin Skywalker.
I think other people have answered it fairly thoroughly, but I think the best answer is really in @ToddWilcox comment to OP. It's not that saving Luke turns him to the good side, but rather the other way around. A telling moment is when Vader is standing behind the Emperor, looking on as the Emperor is killing his son. At one point, he looks at Luke and then at the Emperor, and then back at Luke and back at the Emperor (I forget how many back and forths he does). Even through the expressionless mask, you can see the big WTH going through his head. Like he is thinking "WTH am i doing?" as he finally realizes that evil is evil and has cost him everything and is about to cost him the life of his son. In that moment he turns against the dark side, and demonstrates it by grabbing the Emperor and throwing him over the railing to his death.
You have to remember that Anakin made a pact with The Devil and lost. He was willing to do ANYTHING, even become a Sith, to save Padme and it blew up in his face. Instead of saving Padme as he originally intended, Vader ended up killing Padme, because of Obi-Wan. After having his limbs cut off and being burnt to a crisp and put in a life support suit, Palpatine told Vader "It seems in your anger, you killed her.", which is obviously a lie, because Vader says "She was alive. I felt it!" and before she died, Padme admitted to Obi-Wan that she knew there's still good in Anakin/Vader, meaning Luke wasn't the only one who knew Anakin could still be redeemed and wasn't a lost cause.
Anyway, in Return of The Jedi, Vader realized that Luke, his own son, spared him rather than kill him. That act of kindness made Anakin realize that being Vader isn't worth it anymore and he didn't want the one person he cared about to die or suffer. That's why he sacrificed himself. That's what made Anakin good again and allowed him to become one with The Force.