The Pauli exclusion principle
Only one fermion of a given type is allowed to be in a specific quantum state. A quantum state is a discreet level which can be labeled. The labeling gives information about the spatial characteristics (e.g. the orbit) and the spin of the particle. Two electrons might exist in the same quantum orbital, but they would have to have different spin. No two electrons of the same spin can occupy the same orbital state. This has obvious consequences for atomic structure. For instance, if all the electrons in a Uranium atom could occupy the lowest-possible-energy orbital (which is also the most compact orbital) a heavy atom would be smaller in size than a light atom which is not true.
Sometimes, two states which have different labels have the same energy. In that case it may appear that more than two electrons are in the same state but they are in fact different states. We refer to such states as degenerate. Often this occurs because two particles have the same spin, but can sometimes be practically an accident.