I gate (or Identity gate) is a quantum gate that leaves the state vector unchanged. In other words, it does nothing. It's commonly drawn as a wire. Its matrix representation is equivalent to the identity matrix. For a single qubit, the gate is simply a 2×2 identity matrix.
$$I = \begin{pmatrix}1 & 0\\ 0 & 1\end{pmatrix}$$
You apply the identity gate $I$ to a state vector $\lvert\psi\rangle$ by multiplying it the state vector from the left $I\lvert\psi\rangle$. The following is a worked example where the state vector is a single qubit $\lvert\psi\rangle = a\lvert 0\rangle + b\lvert 1\rangle$
$$I\lvert\psi\rangle = \begin{pmatrix}1 & 0\\ 0 & 1\end{pmatrix}\begin{pmatrix}a\\b\end{pmatrix} = \begin{pmatrix}1a + 0b\\ 0a + 1b\end{pmatrix} = \begin{pmatrix}a\\b\end{pmatrix} = \lvert\psi\rangle$$