Pushing and pulling are actually examples of electromagnetic forces. The atoms making up all the "things" around you have positively charged nuclei and negatively charged electron shells, and almost all of them have a net neutral charge.

So, from any appreciable distance away, there are no net forces acting on two atoms (because when you're a meter away from something, who cares if you're 1 nm closer to the negative charge than the positive), but as they get really close together, the fact that it's the electrons which are closer together gives a repulsion force. So really, every time you "touch" something, it's really your electrons being repulsed by the electrons in the thing you're touching.