Conductor
A conductor is a material, usually a metal, that lets heat or electric current pass through it easily because of its free electrons.
Definition
A conductor is a material, usually a metal, that lets heat or electric current pass through it easily because it has electrons free to move, acting like a wide open road rather than making heat or electricity itself. Metals such as copper, silver, aluminum, and gold are excellent conductors of both heat and electricity, and water with dissolved salts also conducts; the opposite of a conductor is an insulator. In band theory, a conductor is a material whose valence band is partially filled or overlaps the conduction band, leaving electrons free to respond to an electric field, and its electrical conductivity $\sigma$ is the inverse of resistivity $\rho$, so a wire's resistance is $R = \rho L/A$, which rises with temperature in metals as lattice vibrations scatter electrons; the Wiedemann-Franz law, $\kappa/\sigma = LT$, links thermal and electrical conductivity because both are carried by the same free electrons.
Example
Copper wire carries electricity to a lamp, and a metal pot bottom carries heat from the stove into food; a $1\ \text{m}$ copper wire has a resistance of only about $0.02\ \Omega$, so connected across a $1.5\ \text{V}$ battery it would pass $I = V/R = 75\ \text{A}$, far more than the battery can supply, which is why shorting a battery with a wire is dangerous. Copper resistivity is $1.68 \times 10^{-8}\ \Omega\cdot\text{m}$, so a $100\ \text{m}$ wire of diameter $2\ \text{mm}$ (area $3.14 \times 10^{-6}\ \text{m}^2$) has $R = 1.68 \times 10^{-8} \times 100 / 3.14 \times 10^{-6} \approx 0.53\ \Omega$, and carrying $10\ \text{A}$ it dissipates $I^2R = 53\ \text{W}$ as heat along its length.
Key Insight
Pure water is a poor conductor; tap water conducts because of dissolved minerals, and the human body conducts because it is mostly salty water, which is why electricity is dangerous to people. Silver conducts slightly better than copper, but copper wins on cost and aluminum wins on weight for high-voltage lines, so real conductor choice is materials science as much as physics.