The effect of a controlled gate is usually explained as applying the gate in the case the control qubit is in the state |1⟩ and doing nothing when the state of the control qubit is |0⟩. This understanding of controlled gates as classical controls is useful, but has limitations.
An interesting phenomenon occurs when the control qubit is in a superposition of |0⟩ and |1⟩ state and the target qubit is in an eigenstate of an operator U. Let's see what happens when we apply a controlled U gate to the target qubit:
Ctrl U (α|0⟩ + β|1⟩) ⊗ |Ψ⟩ = (α|0⟩ + β exp(2πiθ) |1⟩) ⊗ |Ψ⟩
Here the factor exp(2πiθ) is the eigenvalue of the operator U. What we see in this case is that the state of the target qubit |Ψ⟩ has not changed while a component of the state of the control qubit has acquired a phase which is related to the eigenvalue of the operator U. An important observation to be made here is that this works in a similar manner for both single-qubit or multi-qubit gates U. The phase is given by the eigenvalue of U and can be in principle read out and measured in subsequent steps of a larger algorithm.