Higher-Level is where you make inferences about the population that your subjects are drawn from. It is a bit confusing at times because both (First-Level and Higher-Level) use GLMs, so at both levels you are constructing design matrices, contrasts, etc.
Traditionally, fMRI group analysis has been done in a standard volume space (i.e., Talairach/MNI152/MNI305). With FreeSurfer, we also have the option to analyze group data in the surface space. Volume-based analyses are done in MNI305 space (which is the same as the fsaverage subject).