Plastics are man-made polymers that form flexible substances which can be shaped or moulded. Some synthetic polymers have been designed to serve as electrical conductors and have high conductivity. Most plastics, however, are both thermal and electrical insulators, meaning they show high resistance to electrical current and are poor conductors of heat.
Plastics are generally organic polymers, meaning they consist of carbon, hydrogen and often a couple other elements (chlorine in PVC, oxygen and nitrogen in nylon, and so forth). The long chains of atoms in plastics pack together well to form a flexible solid, so they can be molded into lots of different shapes. Some plastics are thermosetting, meaning you can only mold them once, whereas others are thermoplastics that can be heated then remolded multiple times.
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Metals are excellent conductors of electricity, largely because electrons in the metal are only loosely bound by their respective atoms and thus will readily flow under the influence of an external electric field. Plastics are very different. All the electrons in plastics are tightly bound by their respective molecules and it would take a lot of energy to remove them. Consequently, most plastics are insulators that do not conduct electric current (or do so only poorly).
The same structural features that help make metals good conductors of electricity also make them good conductors of heat -- the electrons are much more free to move and can spread the thermal energy around. In a plastic, by contrast, thermal energy must be transferred by vibrating molecules colliding with each other, so the process is much slower. Most plastics are not good conductors of heat. Making "foam" from plastic can also improve its properties as a thermal insulator. Styrofoam, for example, is an excellent thermal insulator because it contains a great deal of air trapped by the polystyrene.
Many plastics find use as thermal insulators -- in vinyl sidings, for example, or in the Styrofoam beverage cups used to hold hot coffee. Plastic's properties as an electrical insulator also make it useful for wrapping wires -- electrical tape, for instance, is typically made from a kind of plastic called vinyl. In recent years, however, some plastics have been deliberately designed to serve as electrical conductors -- polyacetylene and polypyrrole being just two examples.