Gnunu

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EDIT: Linus, if this is out of place please move it to another thread with Gnunu's comment. I spent way to much time typing this.
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Skip the stuff in parentheses unless you're trying to criticize what I'm saying. I know what I'm talking about, but not everyone can understand higher level concepts at face value, I know I can't it took some good teachers. Everything in parentheses is simply a correction or addition of information to make the following explanation more realistic. I have bolded all information in parentheses so you can easily glaze over it and get to the point at the next sentence (Seriously, skip this shit).
 
"Volts" are a unit used to measure the difference in electrical potential. The measurement for volts that most people are familiar with is actually the difference in voltage between 2 sources of electrical energy (delta V). The potential difference occurs when electrons are held in a higher density in one location relative to another location because the 2 bodies of electrical energy would prefer to share a common energy state (through the higher energy state body releasing electrons to the lower energy state body).
 
A "watt" is a unit of power. Power is the measure the amount of energy changing per second (actually per unit of time to be precise). Here is an equation: P = E/t = (integral of force with respect to time)/t = deltaV*I = (I^2)*R = (V^2)/R = a bunch of other things I can't think of off the top of my head.
 
I will now explain it in a way hopefully most people can understand if you didn't understand voltage or power from the previous statements:
 
A ball is on the ground. You pick up the ball. You drop it. It falls to the ground. What happened?
This is what happened, the ball did not have much energy when it was on the ground, but when you picked it up you gave it the ability to fall. This change in elevation is called potential energy because even though you haven't dropped the ball yet, it has a higher potential to fall. In other words it has more potential to do something (this "something" is work... not important in this scope). This means when you increase the elevation of the ball, the ball has more energy. Think of a higher voltage as raising the ball higher above the ground. Simply put, there is more that can be done.
The force that allows the ball to fall (to do work if you're nitpicky) is one of the 4 fundamental forces. The 4 fundamental forces are gravity, electromagnetism, weak nuclear force, and strong nuclear force.
From the perspective of force/energy, the difference between the ball example and voltage is that the ball drops because of a force generated by gravity, and voltage is similar to a force that's generated by electromagnetism (there would need to be a current - amperage - for there to be a force).
Voltage (the delta V kind, not the absolute kind) is just electrons wanting to move to a place where there are less electrons just like the ball wants to fall to the ground. Just because there is a voltage difference doesn't mean anything is happening just like when you hold the ball its not falling even though it wants to. The potential is there even though nothing is necessarily happening.
 
This leads nicely into the next topic: power
Power (which is measured in the unit called "watt") is the measure of how fast energy changes (to another system(s)... again, topic for another time). Looking back at the example with the ball, power from electricity is similar to how fast the ball falls at a given instance in time (the elevation -aka potential energy- changes at a higher rate the faster the ball falls). The power rating of an electrical component is how many amps it draws at its peak rated voltage capability. P = deltaV*I where P is power, deltaV is voltage, and I is amperage aka current. An amp is the unit for current. Current (when dealing with electrical components) is the rate at which the electrons move through the component (I could have gone deeper down the rabbit hole explaining that an amp is the number of coulombs that move per second and that 1 coulomb equals the magnitude of charge of 1.6*10E19 electrons, but that would distract from the scope of the explanation).
 
Since you mentioned this on a tech website I assume you are referring to overvolting and TDP.
TDP is measured in watts because it refers to how much power the component eats when its operating as hot as it'll go under the worst expected cooling scenario. The reason the component gets hotter when it draws more power is because of the following relationship: Resistance = Voltage/Current. That equation means that as the voltage increases and the current has to increase to keep the resistance constant. Resistance in an electrical component generally won't change in any significant manner unless you heat the component up so much that it starts to kill itself, that's why I said the resistance stays constant. As current increases, more heat is generated by the object the current flows through. You know how when you slowly rub your hands together they don't get hot, but when you move your hand really fast together they get hotter? The way current produces heat is very similar to that. The less current there is, the less the electrons barrage the component they move through. The more current there is, the more the component is effected by the moving electrons. The component expresses this friction like event in the form of heat. This heat is energy that was wasted moving through the component rather than getting through and doing whatever it does to do work before getting to the other end of the system. I'm no expert on computer hardware, but somehow the concept I just described allows your computer components to work harder when you increase maximum power draw capabilities. The only way the components would be able to do this is if somehow the overall resistance dropped. I don't get it when you start going that deep into the operation of the hardware since increasing the max power draw is not overvolting, but power increases and P=V*I so the current has to increase and that forces either the voltage to raise (not overvolting here so this can't be the case) or the resistance has to drop... I have no idea how the resistance would drop.
If overvolting is as simple as it sounds, it is simply the card allowing a larger voltage through itself and therefore allows for more current (and therefore more electrons to do whatever it is they do to make your computer work).
 
Hopefully that helped because I spent way too long writing that.