Right Clicking Deforms instead of moving points? New to blender, 
I just made a simple model, but now when I try to change the mesh, the entire mesh stays together and is moved as well as deforming. For example, attempting to shape a cube by grabbing one vertex no longer pulls that vertex, but moves the entire cube in that direction, and subtly deforms it. I don't have this problem when I 'load factory settings' but every time I load this particular project the problem persists. 

After some googling, it seems like it is acting like I'm deforming a shape key, but I haven't learned much about those yet, and it doesn't look like the mesh even has a basis shape key yet. Does anyone know how to get back to normal editing? And maybe what I accidentally pushed to cause this in the first place?
It sounds like you have proportional editing on.

Try hitting O (NOT zero) on your keyboard and try again.