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In this paper, we classify the singularities of nonnegative solutions tofractional elliptic equation \begin{equation}\label{eq 0.1}  \arraycolsep=1pt \begin{array}{lll}  \displaystyle (-\Delta)^\alpha u=u^p\quad  &{\rm in}\quad \Omega\setminus\{0\},\\[2mm]  \phantom{ (-\Delta)^\alpha }  \displaystyle u=0\quad  &{\rm in}\quad \mathbb{R}^N\setminus\Omega, \end{array} \end{equation} where$p>1$, $\Omega$ is a bounded, $C^2$ domain in $\mathbb{R}^N$ containing theorigin, $N\ge2$ and the fractional Laplacian $(-\Delta)^\alpha$ is defined inthe principle value sense. We obtain that any classical solution $u$ of(\ref{eq 0.1}) is a weak solution of \begin{equation}\label{eq 0.2}  \arraycolsep=1pt \begin{array}{lll}  \displaystyle (-\Delta)^\alpha u=u^p+k\delta_0\quad  &{\rm in}\quad \Omega,\\[2mm]  \phantom{ (-\Delta)^\alpha }  \displaystyle u=0\quad  &{\rm in}\quad \mathbb{R}^N\setminus\Omega \end{array} \end{equation} forsome $k\ge0$, where $\delta_0$ is the Dirac mass at the origin. In particular,when $p\ge \frac{N}{N-2\alpha}$, we have that $k=0$;  when $p< \frac{N}{N-2\alpha}$, $u$ has removable singularity at the origin if$k=0$ and if $k>0$, $u$ satisfies $$\lim_{x\to0}u(x)|x|^{N-2\alpha}=c_{N,\alpha}k,$$ where $c_{N,\alpha}>0$.  Furthermore, when $p\in(1, \frac{N}{N-2\alpha})$, we obtain that there exists$k^*>0$ such that  problem (\ref{eq 0.1}) has at least two positive solutions for $k<k^*$, aunique positive solution for $k=k^*$  and no positive solution for $k>k^*$.

Classification of isolated singularities of nonnegative solutions to fractional semi-linear elliptic equations and the existence results