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1.
Newspeak word, and the most important pillar of Ingsoc. Doublethink is the ability to hold two contradictory opinions at the same time without noticing the contradiction. It also refers to the mind trick to accept the changes to history made by the Party, and in accepting them, forgetting that the mind trick was ever performed, and then forgetting the forgetting, etc.
"Ashes. Not even identifiable ashes. Dust. It does not exist. It never existed."
"But it did exist! It does exist! It exists in memory. I remember it. You remember it."
"I do not remember it," said O'Brien.
by Emperor Spleen June 06, 2004
 
2.
"To know and not to know, to be conscious of complete truthfulness while telling carefully constructed lies, to hold simultaneously two opinions which cancelled out, knowing them to be contradictory and believing in both of them, to use logic against logic, to repudiate morally while laying claim to it, to believe that democracy was impossible and that the Party was the guardian of democracy, to forget, whatever it was necessary to forget, then draw it back into memory again at the moment when it was needed, and then promptly to forger it again, and above all, to apply the same process to the process itself--that was the ultimate subtlety: consciously to induce unconsciousness, and then, once again, to become unconscious of the act of hypnosis you had just performed. Even to understand the word "doublethink" involved the use of doublethink."
-1984, George Orwell
 
3.
When you see potentially creepy and worrisome reasons for someone's words or actions. There are likely plenty of normal explanations, but the creepiness is stuck in your head and you might start to believe it.

You are more likely to doublethink if you are sad, depressed, or recently have experienced something highly disturbing.

Could be caused by a loss of trust in the honesty and innocence of the world, or maybe you're on drugs.
When your boyfriend/girlfriend says "I'll wait for you forever," is it sweet, or creepy?

Michael: "The way Andrew whispers to himself sometimes makes me think he's psycho."
Emma: "Stop double-thinking, he's just trying to remember what the homework was or has a song stuck in his head."

Krystal: "When I found out my boyfriend was a pedophile, I started to doublethink everything my friends and relatives say and do. Maybe they're secretly creepers too."
Jose: "Holy shit. You need help."
by Tmape August 12, 2010
 
4.
When one is forced to express that a lie is truth, despite his knowledge of the real facts. Orgininated in the book 1984 in the fictional Socialist government of Ingsoc
Bobby committed doublethink when he said to his young daughter that she could do anything that she set her mind to it, despite the fact that he knew that was a lie.

(But in the book 1984, doublethink is more of a serious act of course)
by Matthew Peters January 09, 2004