CY applies the Y gate to the target qubit if the control qubit is $|1\rangle$. Like CX, it is an asymmetric Clifford two-qubit gate used for entanglement, though less common than CX or CZ on most quantum platforms.
Matrix (control on qubit 0, target on qubit 1):
$$\text{CY} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & -i \\ 0 & 0 & i & 0 \end{pmatrix}$$
Action on basis states:
$$|00\rangle \to |00\rangle, \quad |01\rangle \to |01\rangle, \quad |10\rangle \to i|11\rangle, \quad |11\rangle \to -i|10\rangle$$
CY can be constructed from CX using phase gates on the target:
$$\text{CY}_{01} = (I \otimes S^\dagger) \text{CX}_{01} (I \otimes S)$$
where $S$ and $S^\dagger$ are phase gates that rotate the Y axis to align with X. Alternatively:
$$\text{CY} = (I \otimes S) \text{CZ} (I \otimes S^\dagger)$$
This shows CY has equivalent power to CX and CZ (all Clifford entangling gates).
CY can be decomposed using single-qubit gates and CX (or CZ):
$$\text{CY} = (I \otimes S^\dagger) \text{CX} (I \otimes S)$$
Gate count: two single-qubit gates (S and S†) and one CX. Useful when CX is more efficient than native CY on a given platform.