@heather Well, just look e.g. at the
controlled NOT - your gate is some sort of unitary operator on the qubit state, and you can apply such unitary operators to
all possible combinations of $\lvert 0\rangle$ and $\lvert 1\rangle$. The operator is full defined by you saying what happens in the cases where the qubit are in either $\lvert 0\rangle$ or $\lvert 1\rangle$ and then extending this linearly.