As @RunarTrollet has pointed out, you need to load the correct packages. My comment had been made upon the following addition to your example code.
\documentclass{article}
\begin{document}
\begin{equation}
\psi= \begin{pmatrix}
[{\overrightarrow{\sigma}\cdot(\overrightarrow{P}-e\overrightarrow{A})+(p_{0}-eA_{0})+m}]\phi \\
[{\overrightarrow{\sigma}\cdot(\overrightarrow{P}-e\overrightarrow{A})+(p_{0}-eA_{0})-m}]\phi \\
\end{pmatrix}
\end{equation}
\end{document}
(Notice that I have added \documentclass{article}, \begin{document} environments, in order to construct a Minimum Working Example.)
Now that I can reproduce your error, I went on to provide this link whose first answer explains the syntax error: \\ followed by [ on the next line will not be interpreted in the way you thought it will do.
However, inserting
\usepackage{amsmath}
before the beginning of the document (before \begin{document}) will change the behavior of \\ to produce your intended matrix notation. For more information on \\, consult here and here.
\usepackage{amsmath}loaded, and it seems to work. You could try loading that package, and see if helps, might be as simple as that. Otherwise, an MWE is a must here. – Runar Jul 06 '16 at 06:00\vec{foo}instead of\overrightarrow{foo}. Also I've added a little bit of whitespace to your code. This was mostly so that it could be hard wrapped and therefore it wouldn't be necessary to scroll to read the code on the site, but you might find it helpful generally for code reability – Au101 Jul 06 '16 at 06:04