How to execute shell script from LaTeX?

I would do something like the following (partially motivated by what Roman suggested): make your LaTeX file be

\documentclass{article}
\begin{document}
\input{scriptoutput.tex}
\end{document}

and generate the file scriptoutput.tex using

/usr/local/bin/my-shell-script.sh > scriptoutput.tex

You could encode this in a makefile if you want to have it run automatically when necessary. Alternatively, you could use the TeX \write18 command,

\documentclass{article}
\immediate\write18{/usr/local/bin/my-shell-script.sh > scriptoutput.tex}
\begin{document}
\input{scriptoutput.tex}
\end{document}

and I think that would automatically run the shell script each time you compile the document. The \immediate is necessary to ensure that the script is run when LaTeX encounters the command, rather than waiting until a page of output is written. (See this question for more on the shipout routine.)

You can do this in TeX. This paper (PDF) shows you how to write and execute a virus within TeX. The same principles apply for executing a shell script. However in my opinion it is more practicable to write a Makefile, which runs before your LaTeX run and inserts the result.

As David pointed out, you can use \write18 to call external programs, then \input the resultant output file. However you will probably want to use \immediate\write18 to make sure the script is executed before calling the \input.

Alternatively, if you use newer versions of pdf(la)tex (after 1.40, I think), you can pipe the output directly into the document, by using a piped input command:

\documentclass{article}
\begin{document}
\input{|"/usr/local/bin/my-shell-script.sh"}
\end{document}

For either method you will need to enable external program calls. For TeXlive distributions, you need to call latex with the -shell-escape option, or for MikTeX, I believe the option is -enable-write18.