Conductors are a category of materials that allow electrons to flow easily (which is called electricity and are a useful way to transport energy). Most conductors are metals, and most metals are conductors, but some metals are better conductors than others. Copper and aluminum are both excellent conductors for making wires because of their high conductivity (low resistivity).
At an atomic level, conductors display little opposition to electrical current; because of this, less energy is required to generate free electrons, implying that the number of electrons in their valence shell is generally fewer than four (they're electron donors). This means that the materials on the left side of the periodic table of elements are metals, and those extra electrons that make them metals allow for electricity to be transmitted with little loss of energy.
Conductors do experience some degree of resistance (conductor resistance), albeit a very small amount. There's a fascinating phenomenon called superconductivity, where resistance actually drops to 0 Ω! This phenomenon was initially discovered in metals, but more advanced superconductors are often made of ceramics. Therefore, not all conductors are metals, but most are.
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