Alignment within cases

With aligned (from amsmath package):

\documentclass[12pt,a4paper]{article}
\usepackage[fleqn]{amsmath}

\begin{document}
\[
ns = \left\{\begin{aligned}\quad
   & s + s + \dotsm + s             & \text{ ($n$ times) if } n & > 0   \\
   & 0                              & \text{ if } n             & = 0   \\
   & (-s) + (-s) + \dotsm + (-s)    & \text{ ($n$ times) if } n & < 0
     \end{aligned}\right.
\]
\end{document}

There should be a wider separation from “(n times)” to “if” than from the formulas and “(n times)”, in my opinion.

The 0 should be left flush, like the other two terms. I also fixed the math error (it should be |n| in the third line).

\documentclass[12pt,a4paper]{article}
\usepackage[fleqn]{amsmath}

\begin{document}

\begin{equation*}
ns =
\begin{cases}
\begin{alignedat}{3}
&s + s + \dots + s          &\quad \text{($n$ times)}   &\qquad& \text{if $n > 0$} \\
&0                          &\quad                      &\qquad& \text{if $n = 0$} \\
&(-s) + (-s) + \dots + (-s) &\quad \text{($|n|$ times)} &\qquad& \text{if $n < 0$}
\end{alignedat}
\end{cases}
\end{equation*}

\end{document}

You can use \hphantom in the second line to move the text over, or alternatively \hfill, as Mico mentions in a comment.

What's the purpose of the \quads anyway?

\documentclass[12pt,a4paper]{article}
\usepackage[fleqn]{amsmath}    
\begin{document}
\begin{equation*}
    ns = 
\begin{cases}
    s + s + \dots + s  & \text{ ($n$ times) if } n > 0 \\ 
   0 & \text{\hphantom{ ($n$ times)} if } n = 0  \\
   (-s) + (-s) + \dots + (-s)  & \text{ ($n$ times) if }  n < 0 
\end{cases}
\end{equation*}
\end{document}

If you want to change the alignment for the columns in cases, one possibility is to define a new environment with the help of mathtools, as described in Right aligned first column in a cases environment For example, to have the first column centered, and the second right aligned :

\documentclass[12pt,a4paper]{article}
\usepackage[fleqn]{mathtools}
\makeatletter
\newcases{crcases}{\quad}{%
  \hfil$\m@th\displaystyle{##}$\hfil}{\hfil$\m@th\displaystyle{##}$}{\lbrace}{.}
\makeatother
\begin{document}
\begin{equation*}
    ns = 
\begin{crcases}
    s + s + \dots + s  & \text{ ($n$ times) if } n > 0 \\ 
   0 & \text{ if } n = 0  \\
   (-s) + (-s) + \dots + (-s)  & \text{ ($n$ times) if }  n < 0 
\end{crcases}
\end{equation*}
\end{document}