The Pauli Matrices are:
\begin{equation}
\sigma_x = \begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}
\qquad
\sigma_y = \begin{pmatrix}
0 & -i \\
i & 0
\end{pmatrix}
\qquad
\sigma_z = \begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix}
\end{equation}
If we start with a state \(\psi\) and act \( X \) on it, the following is the result:
\begin{equation}
X \ket{\psi} = X ( \alpha \ket{0} + \beta \ket{1} ) = \alpha \ket{1} + \beta \ket{0} = \ket{\psi'}
\end{equation}
So \( X \) error is a Bit flip: \( \ket{0} \leftrightarrow \ket{1} \)
If we start with a state \(\psi\) and act \(Z\) on it, the following is the result:
\begin{equation}
Z \ket{\psi} = Z ( \alpha \ket{0} + \beta \ket{1} ) = \alpha \ket{0} - \beta \ket{1} = \ket{\psi''}
\end{equation}
So \( Z \) error is a Phase flip error: \(\ket{0} \rightarrow \ket{0}, \ket{1} \rightarrow - \ket{1} \)