Replacing receptacle, ground and neutral are connected
Hot1 and Neutral1 are the feed from the panel.  Hot2 and Neutral2 feed the next receptacle in the chain.  

If you connect Hot1 and Neutral1 to the receptacle, this receptacle will work.  If you also connect Hot2 and Neutral2 to the receptacle, all the devices downstream will also work. 

Depending on the new receptacle you're installing, you should wire it the same as the old receptacle. 

Just remember that the "black" ungrounded (hot) conductors should be attached to the brass colored screws, and the "white" grounded (neutral) conductors should attach to the silver colored screws.

As for the ground. It's odd that there's only one grounding wire.  Where does it come from, one of the cables, the box, somewhere else?  In any case, the bare grounding conductor should not be connected to the grounded (neutral) conductor.