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...