I am looking for the PostGIS ST_function that will give me the same results as running the ArcGIS Analyis-> Overlays-> Intersect tool on two polygons.
What your looking for is ST_Intersection.
ST_Intersection — (T) Returns a geometry that represents the shared portion of geomA and geomB. The geography implementation does a transform to geometry to do the intersection and then transform back to WGS84.
geometry ST_Intersection( geometry geomA , geometry geomB );
You can see a query that implements it in this question (down at the bottom).