The moon landing is probably the most famous conspiracy theory there is. It happened back in 1969 but there's a lot of dubious evidence given by theorists that it all happened on a film set, along with some conflicting explanations from NASA.
This was all during the Cold War when Russia and America were threatening each other but not actually doing anything. At the very least not doing anything publicly. From there we were given the Space Race, basically a giant, expensive, international pissing contest to see who could do the most space shit first. Background out of the way we move onto the conspiracy theory itself. Russia had already shown they were capable of sending a man into space. America got worried and needed something big to put Russia back in their place. They also needed to be able to prove that it happened so they decided to film it and take a bunch of photographs.
The problem with this was that nobody knew what the moon looked like at that time. At that time. The advances in space exploration that we have today has brought about a number of things that may be evidence it never happened. The most prevalent is this:
In the photo the flag is shown to be rippling. What causes a flag to ripple on Earth? The wind. There's no atmosphere on the moon so there's no way there would be wind. NASA claimed that the rippling was coming from Neil Armstrong planting the flag in the ground and it was just the tremors moving through the pole that caused it to ripple.
The other is that the only source of light is the Sun. If there's only one source off light all of the shadows have to be cast in the same direction but most photos show the astronaut's shadows from a slightly different angle, almost like there was studio lights placed around them, like on a film set.
Theorists claim that the following photo is proof that it was all done on a set:
The reflection in the helmet shows an odd shape suspended in the air. It doesn't make sense that there would be something suspended like that on the moon so people claim it's an overhead light, hanging from a rope. The shape is really hard to make out and bears a vague resemblance to a light. Clutching on straws I think but believe it if you want to.
For all the film buffs out there it's claimed that Stanley Kubrick, director of 2001: A Space Odyssey was hired by the American government to stage the moon landing. Kubrick released 2001 in 1968, one year before the moon landing was supposed to have happened. The film proved that technology was advanced enough to at least stage an authentic representation of space travel. Varying reports say that Kubrick admitted to staging the moon landing on his deathbed and that he places dozens of clues in his other films. My personal favourite example is the room number in The Shining, which is 237. How far away is the Moon from the Earth? 238,000 miles. Holy fuck.
Anyway it probably did happen and people just like to contest each other but you can decide for yourself. There's dozens and dozens of different sites where you can get more information. If you want something factual to decide for yourself have a look at the RationalWiki. If you just want to look at all the crazy shit people come up with then click this bit for fun.
Video: 10 Reasons Why People Believe The Moon Landing Is A Hoax