You may have learned the following statement of the Pauli principle in terms of quantum numbers: electrons can only be in the same orbital if they have different spin quantum numbers, i.e.,

m_s = ± 1/2 .

Quantum chemists use a more rigorous statement that requires more math and a deeper understanding of quantum mechanics: the Pauli principle requires that the wavefunction be antisymmetric. Electrons are fermions and indistinguishable, which means that any pair can be swapped without altering the character of the system. It gets worse from here...