Tikz and Beamer: How to fix coordinate system to be the same on each slide?

you can set the overlay and remember picture option to the tikzpicture:

\documentclass{beamer}
\usepackage{tikz}

\begin{document}
\begin{frame}
\begin{tikzpicture}[overlay,remember picture]
\draw (0,0) circle (1cm);
\end{tikzpicture}
\end{frame}

\begin{frame}
\begin{tikzpicture}[overlay,remember picture]
\draw (0,0) circle (1cm);
\draw (2,2) circle (2cm);
\end{tikzpicture}
\end{frame}

\end{document}

With these options you have also access to the current page anchor, which allows for cool stuff like:

\documentclass{beamer}
\usepackage{tikz}

\begin{document}
\begin{frame}
\begin{tikzpicture}[overlay,remember picture,every node/.style={draw,minimum size=2.5cm,circle}]
\node [anchor=east] at (current page.east){};
\node [anchor=west] at (current page.west){};
\node [anchor=north] at (current page.north){};
\node [anchor=south] at (current page.south){};
\end{tikzpicture}
\end{frame}

\begin{frame}
\begin{tikzpicture}[overlay,remember picture,every node/.style={draw,minimum size=2.5cm,circle}]
\node [anchor=east] at (current page.east){};
\node [anchor=west] at (current page.west){};
\node [anchor=north] at (current page.north){};
\node [anchor=south] at (current page.south){};
\node at (current page.center){};
\end{tikzpicture}
\end{frame}