How do I split a date?

For a simple split function you can use a delimited macro. You could simply do this:

\def\mysplit#1.#2.#3.{\def\myday{#1}\def\mymonth{#2}\def\myyear{#3}}
\def\splitdate#1{\expandafter\mysplit#1.}
\documentclass{article}
\begin{document}
  \noindent We have the date: 23.01.2012 \splitdate{23.01.2012}\\
  Day: \myday\\
  Month: \mymonth\\
  Year: \myyear
\end{document}

result:

Note that right now this approach lacks any kind of error handling for wrongly formatted output.

Edit: Some additional explanation regarding the \expandafter.

The difference is the added \expandafter. What happens when using \splitdate{<argument>} then \mysplit#1 is replaced by \mysplit<argument to \splitdate>. When <argument to \splitdate> is not a date formatted like dd.mm.yyyy but a macro \mydate which is defined as \def\mydate{10.10.2010} then \mysplit\mydate does not match the definition of \mysplit (since that needs arguments delimited by .). When we use \expandafter, \mysplit is left untouched and \mydate is expanded first. This leads to \mysplit10.10.2010, then \mysplit is expanded and the arguments match the definition. When we have an additional layer, like \def\mydate{12.12.2012}\def\otherdate{\mydate} then this will still fail since \expandafter will only expand the token once. We could make this work as well, by adding a local expanded definition which we use. The complete code would then look like this:

\def\mysplit#1.#2.#3.{\gdef\myday{#1}\gdef\mymonth{#2}\gdef\myyear{#3}}
\def\splitdate#1{{\edef\x{#1}\expandafter\mysplit\x.}}
\documentclass{article}
\begin{document}
  \def\mydate{23.01.2012}
  \def\deepdate{\mydate}
  \noindent We have the date: 23.01.2012 \splitdate{\deepdate}\\
  Day: \myday\\
  Month: \mymonth\\
  Year: \myyear
\end{document}

Note that we had to change the definition of \myday etc. to global.

You could also use \StrBefore, \StrBetween, and \StrBehind from the the xstring package:

\documentclass{article}
\usepackage{xstring}

\newcommand*{\ExtractDay}[1]{\StrBefore[1]{#1}{.}}%
\newcommand*{\ExtractMonth}[1]{\StrBetween[1,2]{#1}{.}{.}}%
\newcommand*{\ExtractYear}[1]{\StrBehind[2]{#1}{.}}%

\newcommand*{\MySplit}[1]{%
    \def\MyDay{\ExtractDay{#1}}%
    \def\MyMonth{\ExtractMonth{#1}}%
    \def\MyYear{\ExtractYear{#1}}%
}%

\begin{document}
  \noindent We have the date: 23.01.2012\\
  Day:   \ExtractDay{23.01.2012}\\
  Month: \ExtractMonth{23.01.2012}\\
  Year:  \ExtractYear{23.01.2012}

  \bigskip
  \noindent If you prefer to have separate macros define for each component

  \noindent We have the date: 23.01.2012 \MySplit{23.01.2012}\\
  Day:   \MyDay\\
  Month: \MyMonth\\
  Year:  \MyYear
\end{document}