The common name for a material that allows electricity to flow through it easily is a conductor.
In a conductor, electric charge carriers can move relatively freely. In metals such as copper, aluminum, and silver, those carriers are mainly delocalized electrons. When a voltage creates an electric field, the electrons drift through the material, producing current.
A conductor is not necessarily a perfect conductor: real materials have resistance, so some electrical energy becomes heat. Resistance depends on factors including the material, its length, its cross-sectional area, and temperature. A resistor is deliberately designed to impede current, while an insulator has very few mobile charge carriers.
Current also does not require one particular electron to travel from a battery all the way to a lamp. Instead, an electric field is established through the circuit, and charge carriers throughout the conductor begin drifting. Metals are common conductors, but electrolytes conduct through moving ions, and semiconductors conduct through electrons and holes.