\documentclass{standalone}
\usepackage{tikz}

\begin{document}
\hfil\begin{tikzpicture}

\foreach\i in {0,...,5}{
  \draw(0,\i)--++(5,0);
  \draw(\i,0)--++(0,5);
}
\foreach\i in {0,...,5}{
  \foreach\j in {0,...,5}{
    \fill[color=red](\i,\j)circle(0.1);
    \draw(\i,\j)circle(0.1);
  }
}
\foreach\i in {0,...,4}{
  \foreach\j in {0,...,5}{
    \fill[color=blue](\i+0.5,\j)circle(0.1);
    \draw(\i+0.5,\j)circle(0.1);
  }
}
\foreach\i in {0,...,5}{
  \foreach\j in {0,...,4}{
    \fill[color=blue](\i,\j+0.5)circle(0.1);
    \draw(\i,\j+0.5)circle(0.1);
  }
}

\end{tikzpicture}

\end{document}