Partial correlation is an extension of normal, bivariate correlation. It is useful when you want to look at the relationship between two variables whilst removing or controlling the influence of a 'nuisance' variable. For instance, let's say you test a lot of children and want to see whether their reading ability is correlated with their maths ability (or with the number of lumps in their peanut butter, or whatever). However, a problem here is the nuisance variable of age - it is related to both these variables and so messes up the relationship we want to examine. Partial correlation provides a correlation coefficient with the effects of this extraneous variable removed or 'partialled out', as the jargon goes.

Analyze > correlate > partial...