Materials that are poor conductors of electricity are called insulators. Some examples are solid ionic compounds, non-metal elements, glass and plastics that are polymers of hydrocarbons.
The reason these substances don't conduct electricity is that charges can't move through them. The particles and their electrons are locked into a solid structure.
When ionic compounds are melted or dissolved in water they become conductive because the ions are free to move around, allowing charge to be transferred.
Metals are good conductors because they have mobile valence electrons. The electrons move freely through the solid material in response to a potential difference or charge separation. A metal can be thought of as an arrangement of nuclei in a sea of electrons. When metal is connected in a circuit, electrons flow from where there's more negative charge to where there's more positive charge. If a non-conductor is connected in a circuit this won't happen because the electrons in the non-conductor are associated with a particular atomic nucleus and don't easily leave it.
Some materials, such as silicon, are called semi-conductors because they conduct electricity better than non-metals but not as well as metals.