How to vertically align multiline column

You can try using m{...} type columns (redefining \tabularxcolumn you can change the default p{...} type used by tabularx):

\documentclass[a4paper,12pt,headsepline]{scrartcl}
\usepackage{tabularx}

\renewcommand{\tabularxcolumn}[1]{m{#1}}

\begin{document}

\noindent\begin{tabularx}{\linewidth}{| m{0.25\linewidth}| X |}  
\hline
\LaTeX & Is a document markup language and document preparation system for
the \TeX\ typesetting program.\\
\hline
\end{tabularx}

\end{document}

Please when posting questions always supply a complete document (as in Gonzalo's answer) not a fragment.

Using m is the correct answer to get vertical centering, but one other observation, it may just be an artifact of your small example but in

\begin{tabularx}{\linewidth}{| p{0.25\linewidth}| X |}  

tabularx is going to set the table several times to determine the optimum column width of the second column but since there are no l c or r columns that width is known in advance and you could use

 \begin{tabular}{| m{0.25\linewidth}| m{\dimexpr.75\linewidth-4\tabcolsep-3\arrayrulewidth\relax} |}