How can I place a displaymath environment and an align* environment side by side?

Two possibilities:

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\noindent\begin{tabular}{p{0.45\textwidth}p{0.45\textwidth}}
  \[
    A = B
  \]
  &
  \[
  \begin{aligned}
    a&= b \\
    c&= d
  \end{aligned}
  \]
\end{tabular}

\noindent\begin{minipage}{0.45\textwidth}
  \[
    A = B
  \]
  \end{minipage}
  \begin{minipage}{0.45\textwidth}
  \begin{align*}
    a&= b \\
    c&= d
  \end{align*}
\end{minipage}
\end{document}

Also align* works in a p cell in a tabular environment. A trick to make it work, is grouping the align* environment by curly braces, so the & within align does not act like an & for tabular. Even \\[length] works.

A modification of Torbjørn's table in this way:

\noindent\begin{tabular}{p{0.45\textwidth}p{0.45\textwidth}}
  \[
    A = B
  \]
  &
  {
    \begin{align*}
      a &= b \\[0.5ex]
      c &= d
    \end{align*}
  }
\end{tabular}

It is quite possible you need only:

\documentclass{article}
\usepackage{amsmath}

\begin{document}
\begin{align*}
\begin{aligned}
a &= 1 \\
b &= 1+2 \\
c &= 1+2+3 
\end{aligned}
&&
A = B
\end{align*}
\end{document}

Forget complex tabulars and boxes.