Removing duplicate features with same geometry in ArcMap?

If you have an advanced license, you can use Find Identical or Delete Identical.

Both can be used to find/delete features that have identical attributes, or, if the Shape field is specified, identical geometries.

If you don't have an advanced license, this post will be useful.

In short, you add two fields for X,Y in your attribute table and run a Dissolve on said fields.

If you have polygons which share the same centroid, but are not geometrically identical, you can add two other fields (Perimeter and Area). That should be enough to identify identical geometries for almost all normal cases. See @whuber's comments below.