what type of solids conduct an electrical current?
Solids, and liquids, and gases, that conduct electricity are called Conductors. Although most conductors are metals, not all are, Carbon is not a metal yet under the right conditions it does conduct electricity. Tungsten is a metal, but it doesn't conduct electricity very well, hence it's use in lightbulbs.
Solids, liquids, and gases that do not conduct electricity are called insulators.
And these characterizations are not absolute. Insulators will carry a current if it has sufficient amperage and voltage. A large portion of the electrical energy will be transformed into heat energy, and chemical reactions will probably occur in the insulator, but the current will flow. Conductors will not undergo such changes, nor transform as much energy in to heat, and require far lower voltages and amperages.