what modeling tool is being used to shape the meshes in the linked video demonstration In this animation (from this video) the artist appears to be using a tool that lets him select a vertex, then move the vertex around in such a way that the other adjacent vertices also move, but not as much. The tool appears to create a circle that the user adjusts the size of in some manner.

What tool is he using in this video to shape the spheres?
Press O
 to turn on Proportional Edit in Edit Mode.  Use the mmb 
to scroll the influence range. (or via keyboard with Page Up and Page Down)