This is actually quite a subtle thing that has come up in a few different varieties on the site.
The aligned environment takes an optional argument; in your use case, you have been unlucky enough to select a combination that makes LaTeX think that you are supplying the aligned environment with the argument {\bf Z}, when actually you intend to skip the optional argument altogether, and begin your content with [{\bf Z}].
So, we have to tell LaTeX not to look for an optional argument, which we can do using \relax.
% arara: pdflatex
\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{equation}
\begin{aligned}\relax
[{\bf Z}] &=
\begin{bmatrix}
R_1+j\omega L_1 & -j\omega M \\
j\omega M & -R_2-j\omega L_2
\end{bmatrix} \\
k &=\frac{M}{\sqrt{L_1L_2}}
\end{aligned}
\end{equation}
\end{document}
Alternatively, you can 'protect' [{\bf Z}] by using {[\bf Z]}, which is demonstrated in "[" as first character in table row. However, as discussed in the comments, and in What is the difference between \relax and {}?, \relax is the safer option.
{\bf Z}as a "vector" or "matrix", then consider reading How can I get bold math symbols? – Werner Jun 10 '14 at 04:24