I am using ArcMap 10.6. I created a polygon (red). Some of it overlaps with a feature that delineates a city (green). The brown is where they overlap. In case it makes a difference that the shapefiles are of different GCS's, the green one is North America 1983 and the red one is GCS_WGS_1984. How can I calculate the area of where they overlap? When I Select by Location with my polygon as the target layer and the city as the source layer, and choose Intersect the Source Layer Feature, it just selects the whole polygon.
There are two options (without programming) to do that.
Geoprocessing Tools -> Intersection -> Create new polygon (.shp) which is the result of the overlapping of your shapes. Then you can calculate the area.
Cut the red polygon in two. Start editing -> Cut polygons tool -> Trace (this tool helps you to cut it following the boundaries of the green shape). Then you can calculate the area for the two polygons.
Homologate the projections of your information to avoid future problems