Why do some materials conduct electricity better than others?

That has to do with the ability for electrons to move withing the material. Some metals, like copper, contain a free-moving "lattice" of electrons, which allows a lot of movement. Other materials such as rubber have their electrons very tightly bound, and so have a greater resistance to current.
Some elements do not let go of their electrons as easily as others do. The atoms in a NON-conductive material are 'stable' or balanced while conductive elements pass electrons to each other freely.