These are not really logical classifications, but more like the overall mood of a typeface. So, for example, geometric typefaces can be serif or sans-serif.
-Inspired by the humanist movement that drove the Renaissance
-Driven by the human body, because text was written by scribes, so it had to be efficient for handwriting.
-While driven by handwriting, the open counters also make them very readable on screens
-Examples: Gill Sans, Palatino
-As the name suggests, they are driven by mathematical perfection. Perfect circles, perfect curves, sharp angles.
-Ultimate example is Futura. Another example is Rockwell.
-A middle ground between calligraphic fonts and geometric fonts.
-Examples: Helvetica, Arial, Georgia