Moon landing conspiracy theorists claim that the Apollo program and the moon landings were staged by the U.S. government, specifically NASA. The conspiracy theorists believe that NASA manufactured or tampered with evidence and convinced Neil Armstrong, Buzz Aldrin, and the other Apollo mission astronauts to lie to the public for their entire lives.
As far-fetched as that may sound to the average person — and there is no serious evidence in support of it, and much to contradict it — these conspiracists have been able to maintain broad public interest. In 2001, FOX broadcast a documentary called "Conspiracy Theory: Did We Land on the Moon?" that explored the claim that the U.S. faked the initial moon landing in order to win the "space race" to be the first country to the moon.
A handful of examples from conspiracy theorists on twitter after Neil Armstrong's death Saturday show that these ideas persist, even after high-definition photos taken by the LROC spacecraft have shown the lander modules, tracks left by the astronaut, and the Apollo flags still standing on the moon...
There are many theories that moon landing conspiracists have been touting for decades to "prove" that the manned moon landings were shot on a studio set and that man has still never set foot on the moon. These are the main arguments for their cause:
The Flag Waves
There Aren't Any Stars
There's No Crater Under The Lander
The Crosshairs In The Photos Sometimes Appear To Be Behind Objects
The Shadows Are Inconsistent.
There are hundreds of other ideas conspiracists can give you for proving that the moon landing was faked: from "the pictures are too good to be true" (it was a really nice camera) to "why haven't we gone back?" (being first was that important).
If you're interested in reading more about what the theories are and how they have been debunked, try the super-comprehensive non-profit website Clavius Moon Base.