The rank of Obergrenadier or Oberschütze was one which had no real equivalent in neither the Luftwaffe or the Kriegsmarine. The Waffen-SS, however, had an equivalent, the SS-Oberschütze. The current day US Army equivalent is Private 1st Class.
An Obergrenadier was not a whole lot above his Grenadier platoonmates, at the most commanding a small squad of 5 or 10. It was usual for weapons in limited supply, like Panzerfausts, to be carried by an Obergrenadier.
For a more complete overview of which ranks existed, see Wehrmacht ranks.
The page that helped me with this writeup is no longer on the internet, but it used to live at www.wehrmacht-association.com/Open_Pages/ranks.htm