In this paper, we study the problem for a nonlinear elliptic system involving fractional Laplacion: $$ \begin{cases} \ (-\Delta)^{\frac{\alpha}{2}}u=|x|^{\gamma}u^{p}v^{q+1},\\ \ (-\Delta)^{\frac{\beta}{2}}v=|x|^{\tau}u^{p+1}v^{q}, \end{cases} $$ where $0<\alpha,\beta<2,$ $p,q>0\ and\ \max\{p,q\}\geq1,$ $\alpha+\gamma>0,\beta+\tau>0,$ $n\geq2$. First of all, while in the subcritical case, i.e. $n+\alpha+\gamma-p(n-\alpha)-(q+1)(n-\beta)>0$, $n+\beta+\tau-(p+1)(n-\alpha)-q(n-\beta)>0$, we prove the nonexistence of positive solution for the above system in $\mathbb{R}^{n}$. Moreover, though $Doubling\ Lemma$ to obtain the singularity estimates of the positive solution on bounded domain $\Omega$. In addition, while in the critical case, i.e. $n+\alpha+\gamma-p(n-\alpha)-(q+1)(n-\beta)=0$, $n+\beta+\tau-(p+1)(n-\alpha)-q(n-\beta)=0$, we show that the positive solution of above system are radical symmetric and decreasing about some point by using the method of $Moving\ planes$ in $\mathbb{R}^{n}$.