Bold text and newlines inside equations

You could bold an equation with \boldsymbol. Use \align for multiline equation with alignment.

\begin{align}
\boldsymbol{w_{n+1}} &= \boldsymbol{w_n} + e\eta\boldsymbol{x}(n) \\
\boldsymbol{\Delta w} &= e\eta\boldsymbol{x}
\end{align}

\textbf is for text mode, not math mode. Also, inline math ($ ... $) is for use inline, and so does not produce new lines. This is pretty basic math stuff for TeX: perhaps you should read something like Math Mode by Herbert Voss (type texdoc mathmode at the Terminal/Command Line).

I think you would be best using the align environment:

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{align}
  \mathbf{w_{n+1}}  &= \mathbf{w_{n}} + e\eta\mathbf{x}(n) \\
  \mathbf{\Delta w} &= e\eta\mathbf{x}
\end{align}
\end{document}

Use $\mathbf{w_{n+1}}$. However, if you want bold symbols, you'd have to use $\pmb{\eta}$