Change figure numbering for appendix

Redefine \thefigure when your appendices start, and set the figure counter to zero at the beginning of each appendix.

\documentclass{article}
\begin{document}
\begin{figure}
\centering\rule{1cm}{1cm}
\caption{This is a figure}
\end{figure}
\appendix
\renewcommand\thefigure{\thesection.\arabic{figure}}    
\section{A nice appendix}
\setcounter{figure}{0}    
\begin{figure}
\centering\rule{1cm}{1cm}
\caption{This is a figure in appendix A}
\end{figure}
\end{document}

You can use the chngcntr package which includes the command \counterwithin.

Using this as \counterwithin{figure}{section} changes the figure numbering from that point on so that the section number is included and resets the numbering of figures at the beginning each subsequent section. An example is:

\documentclass{article}

\usepackage{chngcntr}

\begin{document}

\section{Introduction}

\subsection{Problem description}

\begin{figure}[htp]
  \centering Figure
  \caption{Call this figure 1.}
  \label{fig1}
\end{figure}

\appendix
\counterwithin{figure}{section}

\section{Figures}

\begin{figure}[htp]
  \centering Figure
  \caption{Call this figure A.1.}
  \label{figa1}
\end{figure}

\begin{figure}[htp]
  \centering Figure
  \caption{Call this figure A.2.}
  \label{figa2}
\end{figure}

\end{document}

A simple way to prepend an A to your appendix figures and to have the counter reset is to paste the following two lines before your first figure of the appendix.

\renewcommand{\thefigure}{A\arabic{figure}}

\setcounter{figure}{0}

as illustrated here.