What does "genetic correlations" mean as opposed to just correlations?
It is simply a correlation where one of the two variables considered is the genetic state at a given locus. In a GWAS, one loops through all loci and systematically performs a correlation analysis between the genetic states at this locus and the phenotypic trait of interest.

In the example of sentence you cite


  Genetic correlations of same-sex sexual behavior with various preselected traits


They looped through each polymorphic locus in their data set and at each locus they performed a correlation analysis between the two variables "genetic states at a this locus" and "same-sex sexual behavior" (whichever way this second variable is recorded).

A study might also be interested in more than one phenotypes. In such case, different phenotypic variables can either be considered separately or can be included in the same analysis through linear models. It is for example not so unusual to include one of the variable as a covariatiate in an ANCOVA type of design.