Metals conduct electricity better than any other kind of material. The reason electrons flow so easily in metals is because most of them are free. That is to say, the electrons don't feel bound to their original host, and can float around an ocean of metal atoms. In a typical metal, there's about one electron per atom that is free to move about. Without an electric field or electric force applied to the metal, these free electrons move around randomly and there the current is in no particular direction. Once an electrical force is applied, the electrons will move in the opposite direction the force is applied.

Electrical conductivity refers to how easily electrons can move around in a material. The units of electrical conductivity are Siemens per meter (a Siemen is just an inverse ohm, which is the unit for electrical resistance). In my line of work I measure electrical conductivity in units of Siemens per centimeter, but it's the same difference, inverse resistance per unit distance.

You can measure the electrical conductivity by measuring the electrical resistance. Take whatever material you want to study and hook it up to a voltage source, like a battery. Measure the current flowing through your sample as a function of voltage using an ammeter. Ohm's law states that the resistance is proportional to voltage and inversely proportional to current
R = V/I
(with resistance R, voltage V, and current I).

What you'll likely find after plotting your data is that the current scales linearly with voltage, so the inverse slope of your line gives you your material's resistance. With resistance in hand, you can calculate resistivity. Resistivity is equal to the resistance multiplied by the cross-sectional area of the sample and divided by its length (resistivity = R*A/l with resistance R, cross-sectional area A, and length l). Conductivity is simply the inverse of the resistivity (conductivity = 1/resistivity).

That's the essence of how the conductivity of pure metals is measured. Electrical contacts are connected between the metal and an electrical power source and multimeter, and the resistance is calculated by measuring the current as a function of voltage. The resistance is then used with the dimensions of the sample to calculate resistivity and conductivity.

Conductivity σ is generally measured in units of Siemens per meter (S/m). Siemens is a measure of conductance G, and is the mathematical reciprocal of resistance (units of Ohm,Ω). Rarely, Siemens can also be called “mho” (the lettering inverse of “ohm”). Macroscopic measurements of objects like metals can be fairly straightforward. One can connect a voltage or potential difference (units of Volt, V) across a piece of, say, metal wire, and measure the resulting current (units of Amp, A). Using Ohm’s Law, IR = V, where I is the current and V is the voltage, we can calculate the resistance R. Then, taking the reciprocal of this value gives us the conductance G. If we know the length x of the wire, we can divide the conductance by this length G/x to obtain the desired conductivity in Siemens per meter. Alternatively, one can apply an electric field E (units of Volts per meter, V/m) and measure the resulting current I. Then, if we know the cross-section area A of the wire, we can once again derive conductivity using the equation σ = I/AE.

Electrical conductivity works by having electrons that are free to move around in the material. When a voltage is applied, the negatively charged electrons will tend to drift towards the positive field. Scientists measure the electrical conductivity in units of siemens per metre (S/m). However, what scientists actually measure is the electrical resistivity, which is the opposite. (Something that is more conductive is less resistive.) The units for electrical resistivity are ohm metres (Ohm.m). The way that we measure the electrical resistivity is by applying a potential to two probes, and then measuring the electrical resistance between those probes. For example, when you have a car battery. These probes can be anything that conducts electricity, so you could just have a couple pieces of copper wire, a power source, and an ohmmeter, which is a device to measure the resistance.