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 discrete 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 be required 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 different states have have the same energy. We refer to such states as degenerate. Often this occurs because two particles have the same spin, but can sometimes be practically an accident.