What are examples of coherent sheaves $\mathfrak{F}$ on a compact complex $n$fold with $\dim \operatorname{Supp} \mathfrak{F}=p$ , where $0\leq p \leq n$ ? And how can they be described in local coordinates under the equivalence with vector bundles?

Coherent sheaves are much more general than vector bundles. A vector bundle corresponds to a locally free sheaf; the support of such a sheaf is the whole variety. If $X$ is your variety, of dimension $n$, and $Y \subset X$ is some closed subvariety of $X$, and $\mathcal{F}$ is some sheaf on $Y$, then there is a coherent sheaf $\mathcal{G}$ on $X$ whose sections over an open set $U$ are the sections of $Y$ on $U \cap Y$ (this is the "extension of $\mathcal{F}$ by 0"). If $\mathcal{F}$ is locally free (e.g. if $\mathcal{F}$ is the structure sheaf of $Y$ itself) then the support of $\mathcal{G}$ is precisely $Y$. This gives a nice supply of coherent sheaves whose support is easy to understand. 


Take an affine patch $S\subseteq X$, then $S\hookrightarrow\mathbb{A}^N\subseteq \mathbb{P}^N$ for some $N\geq n$. Now take a (generic) hyperplane section of $S$ in $\mathbb{P}^n$, this will have dimension $n1$. Let $Y$ be the closure of this in $X$, then $Y$ is a codimension 1 closed subvariety of $X$. Reapplying the process enough times to $Y$ itself will give you a closed $p$dimensional subvariety $Z$ of $X$. The ideal sheaf $\mathcal{I}_Z$ is then a coherent sheaf on $X$, and the sheaf $\mathcal{O}_Z:=\mathcal{O}_X/\mathcal{I}_Z$ is coherent and supported on $Z$. Note that this method cannot be applied to general complex analytic varieties. 

