Partial correlation is by definition the unique correlation between two variables. In Venn diagrams it is the overlap between two circles without the overlap of the others.
Suppose three variables (all measured at an interval or ratio scale) are used in a research. Between these variables correlations can be computed. If there is no correlations between two variables (that is rx1x2 = 0), a Venn diagram will show no overlap. If there is a perfect correlation between two variables (that is rx1x2 = 1), a Venn diagram will show a complete overlap. A correlation between 0 and 1 will show a partial overlap.
Between 3 variables some overlap will be seen between all three variables. Some part will be unique. The unique part is the partial correlation coefficient. As a matter of fact three partial coefficients can be computed. The notation of the partial correlation is a small r with subscripts like r12.3. Notice the dot between 12 and 3. This means the correlation is computed between variable x1 and x2 controlled for x3. Now three different types of partial correlation can be computed: r12.3, r13.2 and r23.1.
The usefulness of partial correlations becomes clear in regression analysis. In these analysis partial correlations are calculated for adding a predictive variable or a group of predictive variables.