First, I don't understand what it means by partial correlation matrix. I know correlation matrix, though.

Second, assuming partial correlation matrix is similar to correlation matrix, then how can you refer the connections (edges) between the dots (genes) and how to resolve the directionality?

Third, I am using the R package "parcor". I have read the R manual, some reference papers, but still don't know how to infer a network from the package. Can somebody give a simple tutorial?

Thanks much in advance.
The partial correlation between X and Y is the correlation between the residuals of X and Y after linear regression to eliminate the effect of confounding factors. For example, assume you have two independent variables W and X and a dependent variable Y. To compute the partial correlation between X and Y, you regress X on W to get the residual rX and regress Y on W to get residual rY. The partial correlation is then the correlation between rX and rY. It allows to answer questions like how much X contributes to Y when the effect of W is removed. 
As the partial correlation can be viewed as a similarity measure, a partial correlation matrix can be seen as the adjacency matrix of a graph where the partial correlations represent edge weights. Since it's a correlation matrix, it is symmetric. A symmetric adjacency matrix implies an undirected graph so you need additional information to build a directed graph out of it.  
As for the practical aspect, once you have your matrix, you can use the [graph_from_adjacency_matrix()][1] function from the [igraph package][2] to get the coresponding graph structure.


  [1]: http://igraph.org/r/doc/graph_from_adjacency_matrix.html
  [2]: http://igraph.org/r/