How to justify an element in matrix

I would use a cases environment (from the amsmath package):

\documentclass[10pt]{article}
\usepackage{amsmath}

\begin{document}

\[
{{R}_{1}}( {{x}_{pi}},{{G}_{q}},{{x}_{qj}} ) = 
\begin{cases}
   +\infty  & p=q  \\
   \sum\limits_{l=1}^{d}\,({{x}_{pi}}[ l ]-{{x}_{qj}} [ l ])\bigl( 2( {{x}_{qj}} [ l ]-{{{\bar{x}}}_{q}} [ l] \bigr)+({{x}_{pi}}[ l ]-{{x}_{qj}} [ l])(\lvert {{G}_{q}} \rvert|-1)/\lvert{{G}_{q}}\rvert ) & p\ne q  \\
\end{cases}
\]

\end{document}

I removed some superfluous \left, \right constructs and replaced one of them with \bigl, \bigr.

You can use an array instead, and then indicate the column specification. For example, using @{}lc produces a similar layout to your matrix yet left-aligned:

\documentclass{article}
\usepackage{amsmath}% http://ctan.org/pkg/amsmath
\begin{document}
\[
{{R}_{1}}\left( {{x}_{pi}},{{G}_{q}},{{x}_{qj}} \right)=\,\left\{ \begin{matrix}
   +\infty  & p=q  \\
   \underset{l=1}{\overset{d}{\mathop \sum }}\,({{x}_{pi}}\left[ l \right]-{{x}_{qj}}\left[ l \right])\left( 2\left( {{x}_{qj}}\left[ l \right]-{{{\bar{x}}}_{q}}\left[ l \right] \right)+({{x}_{pi}}\left[ l \right]-{{x}_{qj}}\left[ l \right])(\left| {{G}_{q}} \right|-1)/|{{G}_{q}}| \right) & p\ne q  \\
\end{matrix} \right.
\]

\[
{{R}_{1}}({{x}_{pi}},{{G}_{q}},{{x}_{qj}})=\left\{ \begin{array}{@{}lc}
   +\infty  & p=q  \\
   \sum\limits_{l=1}^d\,({{x}_{pi}}[ l ]-{{x}_{qj}}[ l ])( 2( {{x}_{qj}}[ l ]-{{{\bar{x}}}_{q}}[ l ] )+({{x}_{pi}}[ l ]-{{x}_{qj}}[ l ])(| {{G}_{q}} |-1)/|{{G}_{q}}| ) & p\ne q  \\
\end{array} \right.
\]

\end{document}

The excessive use of \left and \right is not required. It introduces unwanted spacing and doesn't add any value. Consider using less-invasive constructions like \bigl and \bigr.