This is easiest to do with the enumitem package. You can either specify topsep=0pt or probably better nosep which will also reduce the vertical spacing between items:

\documentclass{article}
\usepackage[shortlabels]{enumitem}
\begin{document}
This is first line and i need to remove space between this line and
item list below it.
\begin{enumerate}[-,topsep=0pt]
\item asdasd
\item dsfdsf
\end{enumerate}
This is first line and i need to remove space between this line and
item list below it.
\begin{enumerate}[-,nosep]
\item asdasd
\item dsfdsf
\end{enumerate}
\end{document}
To set such an option globally, use \setlist:
\documentclass{article}
\usepackage[shortlabels]{enumitem}
\setlist[enumerate]{nosep}
\begin{document}
This is first line and i need to remove space between this line and
item list below it.
\begin{enumerate}[-]
\item asdasd
\item dsfdsf
\end{enumerate}
\end{document}
enumitem with the shortlabels option essentially reproduces the behaviour of the enumerate package.
--is better. – egreg Dec 06 '13 at 10:05