Is Darth Vader really Luke's father?

Yes. In the original trilogy, Luke finds out when he is in the middle of a conversation with Vader and as Luke claims Vader killed his father, Vader shouts, “Luke! I am your father!”. At this point, Luke shakes his head and says ,”No!”.

In the new trilogy, the last film Revenge of the Sith, Anakin is tricked by the Lord Sith by having Anakin’s psychic visions foresee that Padmé Amidala, his wife, dies during birth. With this, he sets out to kill everyone, but Obi-Wan fights and defeats him by having Anakin sink in a bad of lava. After he is recovered, he is robotically reborn as Darth Vader, a title given for the Dark Side followers.

Meanwhile, Padmé Amidala gives birth to twins, Luke Skywalker and a sister who is adopted by Bail Orgena and given the name Leia Orgena.

Yep, small cosmos eh?

Common mistake. He says “No, I am your father” He doesn’t say Luke. :)

That explains things. I’m always confused who Anakin and Luke Skywalker are

Yep.

yah, its only like the biggest line in the movie (and book), which they use in like all other movies now. example: Toy story two, Psych… so yes.