You can't put a floating object inside a minipage, but you can place minipages inside a floating object. I notice that you are using the obsolete subfigure package; you should use subfig or subcaption instead.
Below there are two approaches; the first one using the subfig package, and the other one using the subcaption package:
\documentclass{article}
\usepackage[demo]{graphicx}
\usepackage{caption}
\usepackage{subfig} % Needs subfig for this example.
\begin{document}
\begin{figure}
\begin{minipage}{.48\textwidth}
\centering
\subfloat[]{\includegraphics[width=.4\linewidth]{fig1}}\quad
\subfloat[]{\includegraphics[width=.4\linewidth]{fig2}}
\caption{First figure with two subfigures}
\label{fig:testa}
\end{minipage}\hfill
\begin{minipage}{.48\textwidth}
\centering
\subfloat[]{\includegraphics[width=.4\linewidth]{fig3}}\quad
\subfloat[]{\includegraphics[width=.4\linewidth]{fig4}}
\caption{Second figure with two subfigures}
\label{fig:testb}
\end{minipage}
\end{figure}
\end{document}

\documentclass{article}
\usepackage[demo]{graphicx}
\usepackage{caption}
\usepackage{subcaption}
\begin{document}
\begin{figure}
\begin{minipage}{.48\textwidth}
\centering
\subcaptionbox{}{\includegraphics[width=.4\linewidth]{fig1}}\quad
\subcaptionbox{}{\includegraphics[width=.4\linewidth]{fig2}}
\caption{First figure with two subfigures}
\label{fig:testa}
\end{minipage}\hfill
\begin{minipage}{.48\textwidth}
\centering
\subcaptionbox{}{\includegraphics[width=.4\linewidth]{fig3}}\quad
\subcaptionbox{}{\includegraphics[width=.4\linewidth]{fig4}}
\caption{Second figure with two subfigures}
\label{fig:testb}
\end{minipage}
\end{figure}
\end{document}

The demo option for graphicx simply replaces actual figures with black rectangles; do not use that option in your actual document.