So you understand more of what happens when a photographer develops your photographs, or "processes" them, I offer this brief video. Think of developing and processing as pretty much synonymous except that some photogs may use "processing" when they mean changing the photo in Adobe Photoshop or a similar application. I always refer to the process through which I put all your photos before delivery as "developing". That's what I've called it since it meant dipping film in chemicals to cause the picture to appear.
So developing is how I correct the colors after shooting in varied lighting, how I improve the exposure after shooting a parade at night, in general making sure the scene I shot looks as good as possible. Photoshop is used sometimes to do the same but more often to change the composition of the photograph, like removing an unwanted object.