Start an edit session and digitize a polygon around the area you would like removed. After the new polygon is created and is selected click the Editor dropdown > Clip. Keep the defaults on the Clip dialog and click OK:
Then delete the polygon you created and the area underneath should be removed.
You can also use the cut polygon tool in your editor bar. Select the polygon you want a piece cut out of, click on "Cut Polygons Tool", draw around the piece that needs to be gone, double click once your box is drawn. Once it is cut out select it and delete it.
That's what I love about ArcGIS so many ways you can do things 🙂