How to georeference a shapefile in ArcGIS / Calibrate feature in ArcMap
Georeferencing shapefile using ArcGIS for Desktop
You want to use the "Spatial Adjustment" toolbar in ArcGIS, it works much like the georeferencing tool, but for vector data rather than raster. It can be applied to an entire layer at once, so you can create just a few control points (they're called "displacement links" in the spatial adjustment tool) and move every polygon in your layer.