Performing a Spatial Join will do this. Right click on the point layer and choose "Joins and Relates > "Join". In the Join Data dialog box, choose "Join data from another layer based on spatial location" in the drop down. Then choose the polygon layer you want joined. Then choose the radio button that says "is closest to it". (The selections are a little different if you're joining points to lines)
This function is also available in ArcToolbox, which is supposed to give better performance with large datasets, and provides some extra functionality.
This method is available in all versions of ArcGIS, and at all license levels.