Take an open neighborhood $U$ of $p\in X$ which is a closed complex subvariety of a domain $D\subset \mathbb C^m$ with coordinates $z_1,\dots,z_m$.

Let $f_1,\dots,f_k$ be functions on $D$ such that $O_{X,p}=O_{D,p}/(f_{1p},\dots,f_{kp})$, where $f_{ip}$ denotes the germ of $f_i$ at $p\in D$.

We denote by $J_p(f_1,\dots,f_k)=\left(\frac{\partial f_i}{\partial z_j}(p)\right)$ the Jacobian matrix of the $f_i$ at $p$.

Then $rank J_p(f_1,\dots,f_k)+embdim_pX=m$, where $embdim_pX$ is the embding dimension of $X$ at $p\in X$.

What about some example to verify the above result?

you? See the comments on your older, closed question – Yemon Choi Apr 25 '13 at 7:41