Pauli gates are one of the three quantum gates $(X, Y, Z)$, which correspond to Pauli matrices $(\sigma_x, \sigma_y, \sigma_z)$, with the inclusion of the identity gate $I$.

For a single qubit, they take the following matrix form $$I = \begin{pmatrix}1 & 0\\0 & 1\end{pmatrix}\quad X = \begin{pmatrix}0 & 1\\ 1 & 0\end{pmatrix}\quad Y = \begin{pmatrix}0 & i \\ -i & 0 \end{pmatrix}\quad Z = \begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}$$

They are involutory, meaning they square up to identity matrix $$I^2 = X^2 = Y^2 = Z^2 = -iXYZ = I$$

Multiplying two gates produces the third gate with an induced global phase shift of $\pi/2$ radians (because $e^{i\pi / 2} = i$) $$XY = iZ \qquad YZ = iX \qquad ZX = iY$$

The Pauli gates $(X, Y, Z)$ anticommute, meaning multiplying them in revrse order produces a minus sign (equivalent to a global phase shift of $\pi$ radians, since $e^{i\pi} = -1$). $$XY = -YX\qquad YZ = -ZY\qquad ZX = -XZ$$