Position a node a certain distance away from another node at a certain angle in TikZ?

Yes, TikZ can use polar coordinates. Not sure exactly what is the best way of defining the second node, here is one way.

\documentclass{scrartcl}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}
  \node (A) {A};
  \path (A) ++(30:2) node (B) [draw,fill=blue!20] {B};
\end{tikzpicture}
\end{document}

In (30:2) 30 is the angle, 2 the distance. 0° is to the right, positive direction is counterclockwise, so node B will be above and to the right of A.

TikZ allows you to specify positions and vectors in polar coordinates. Here's an example:

\begin{tikzpicture}
\node at (0,0) (a) {A node};
\node at (120:1cm) (b) {B node};
\path (a) ++(-45:2cm) node (c) {C node}; 
\end{tikzpicture}

You can shift node position B regarding to another node A from an angle and a radius in the following way:

\usepackage{tikz}
\begin{document}
 \begin{tikzpicture}
   \node (A) {A};
   \node (B) at ([shift=({240:1 cm})]A) {B};
 \end{tikzpicture}
\end{document}