CNOT is the most common two-qubit gate: flips the target qubit if the control qubit is $|1\rangle$. Essential for creating entanglement.
Matrix (control on qubit 0, target on qubit 1):
$$\text{CNOT} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{pmatrix}$$
Action on basis states:
$$|00\rangle \to |00\rangle, \quad |01\rangle \to |01\rangle, \quad |10\rangle \to |11\rangle, \quad |11\rangle \to |10\rangle$$
CNOT creates Bell (maximally entangled) states:
$$\text{CNOT}(H \otimes I)|\psi\rangle = \frac{|00\rangle + |11\rangle}{\sqrt{2}} \quad \text{(if } |\psi\rangle = |00\rangle\text{)}$$