Since we live in a world of standards the thickness of a wire is called it's gauge, more specifically the American Wire Gauge (AWG). The smaller a wire's AWG, the larger the wire's diameter. Typical wires are 12, 14, 16, 18, and 20AWG, but this goes all the way down to the 4/0 (- 4AWG) wire used in power plants and up to the 40AWG wire used inside earbuds. A chart exists (on my site
A damn should given because wire is not perfect; it has some resistance. Resistance is the opposition to current flow in a conductor, and when there is resistance there is also heat. This is because some voltage is lost in a conductor with resistance, and when voltage is lost power is lost. That power can't just disappear, so it turns into heat.
The thinner a wire is the more resistive it is due to the smaller cross sectional area of the wire. With a smaller CSA there are less atoms available for electrons to jump around on, and that makes the wire more resistive. The current-carrying limit of a wire is called it's ampacity, and there is even a chart
for that. Problem is, aside from house electricians nobody uses that chart as it's very conservatively rated. Real men calculate acceptable ampacity themselves using a resistivity table.
Using ohm's law one can figure out the power dissipated in the wire by using the formula above. Power is W = E * I2, so if we had to push 15 amps through 25 feet of 14ga extension cord, we'd dissipate 0.06313ohms * 15amps2 = 14.2watts one way through the cord, and another 14.2W on the return path. A total of 28.4 watts would be lost as heat through the cord. It'll get a bit warm, but it won't melt.
There's another issue with wire resistance we have to be concerned about and that is voltage loss through the wire. As you now know, pushing more current through a wire will dissipate more power, and when you dissipate power you are also dissipating voltage. In order to calculate how much voltage is lost in a length of wire you find the wire's resistance, then multiply that by current (E = I * R). Using our 25 foot extension cord, we'd find that pushing 15A through that would cause 0.94V to be dropped one way. This leads to a total voltage loss of 1.88V since the electricity has to return through the other wire in the cord as well.