Step 1: Understanding Pauli's exclusion principle.
Pauli's exclusion principle states that no two electrons in an atom can have the same set of quantum numbers. This leads to the fact that each orbital can hold a maximum of two electrons, and they must have opposite spins.
Step 2: Explanation.
Since there are four quantum numbers and two electrons in an orbital, one electron must have a spin of +1/2, and the other must have a spin of -1/2.
Step 3: Conclusion.
Thus, Pauli's exclusion principle explains why an orbital can hold only two electrons.