Brain in a vat
A brain in a vat is the idea that a mind can exist in a situation in which it experiences an entirely simulated reality. The person thinks, feels and lives their life completely unaware that they do not experience the outer reality.
One of the earliest and most influential expressions of the idea was Plato's allegory of the cave. He imagined that people were restrained in a cave and had only ever seen one wall in front of them. On this wall are shadows cast by a light source behind them. People walk past the cave mouth and the people trapped in the cave might believe that the shadows were real. Plato's intention was to explain idealism rather than a brain in vat. The difference between them is that the shadows Plato's cave has indirect resemblance to the true world. A brain in a vat could, in principle, experience a very different reality.
In 1641, René Descartes speculated in his Meditations on First Philosophy that the world might be an illusion created by an evil demon. Ultimately Descartes rejected the idea that this could possibly be true, but in order to do this he had to invoke the idea of a good God who would not allow this to happen. This line of reasoning is not open to atheists.
Indeed, it seems that there is no way to conclusively prove that the world is not an illusion, if we are allowed to make the assumption that such an illusion would be so skillfully made that it was indistinguishable from reality. This may lead to a position of an extreme skepticism in which knowledge is impossible. Another consequence is that science is only able to state facts about the world with a limited degree of certainty.
If we cannot even assert conclusively that the real world exists, then it is fair to say that we can never truly "know" whether something like a god exists. Therefore, the brain in a vat is sometimes brought up as a way of illustrating why weak atheism makes more sense than strong atheism.
Do we live in a computer simulation?
Some cosmologists have speculated that since there may be many more simulated universes than real ones, it is quite likely that we are currently in a simulated reality. Theoretical physicists are trying to think of ways of testing this hypothesis, such as artefacts or glitches in reality caused by limitations in the simulation, but so far no evidence has been found to support the idea.
In popular culture
The idea of a brain in a vat has been used in various TV shows and movies. By far the most well known movie to use the idea is The Matrix (1999). Other usages including: Red Dwarf (episode "Back to Reality", 1992), Star Trek the Next Generation (episode "Ship in a Bottle", 1993), eXistenZ (1999), The Thirteenth Floor (1999), Source Code (2011) and in many other places 
"Have you ever had a dream Neo, that you were so sure was real? What if you were unable to wake from that dream? How would you know the difference between the dream world and the real world?"
- — Morpheus in The Matrix
"Is this the real life? Is this just fantasy?"
- — Queen, Bohemian Rhapsody