How to understand electron drift velocity? I know that electron drift velocity can be defined as $I$ = $nAve$, where $I$ is the electric current, $A$ is the cross-sectional area of a conductor, $v$ is the electron drift velocity in question, and $e$ is the fundamental charge.
But how does the electron drift velocity affect an electric circuit? Does it determine the speed of electricity? Why is the speed of electricity different from the velocity of propagation of electrons?
 A: Here is an anlaogy to get the flavor of the idea. Suppose you want to transport energy from A to B. So you get a big spring and lay it between A and B. You hit the end of the spring with a hammer to inject some energy. A compression pulse travels down the spring and kicks a receiver at the other end, delivering the energy. Suppose every time you hit the spring, it moves a millimeter.
The speed of the pulse is like the speed of electricity. The speed of the spring is like drift velocity.
The speed of the pulse depends on the properties of the spring. A stiff spring has strong forces between the coils. A pulse travels faster in a stiff spring than a Slinky. Likewise in a light spring, the forces accelerate a small mass to a high velocity.

The spring analogy works well with electrons. Normally there are the same number of electrons and protons in a wire. But it is possible to "hammer" some extra electrons into one end. The electrons repel each other. It takes energy to crowd them together, doing work against the repulsive forces. When you do crowd them together, the excess negative charge generates an electric field.
The repulsive forces cause a pulse of electrons to travel down the wire. The electrons don't move very far. They quickly run into more electrons and crowd them. The original electrons come to a stop and the new electrons carry the pulse. This is like the pulse in a spring moving from coil to coil.
When the pulse arrives at the other end, enough electrons pop out into the receiver to deliver the energy and bring the excess charge in the wire down to $0$.
The electric forces between electrons are extremely strong. It takes very few electrons to carry the energy. The excess charge and resulting electric field are usually not noticeable. Also an electron is very light, about $1/1800^{th}$ the mass of a proton. This combination makes electric pulses travel extremely fast. In copper, it is about $1/3$ the speed of light.
Every time you inject a pulse, all the electrons in the wire move over a few electrons. Not very far. Just enough to make room for a few incoming electrons and to eject a few electrons on the other end.
If you continually inject electrons, they would slowly move down the wire. But the speed at which the crowding propagates down the wire is much faster. When an electron is pushed in, they all shuffle a bit, and almost immediately an electron is pushed out the other end.

As an aside, people came up with names like "potential difference" and "electromotive force" in the 1800's. This was before the existence of atoms was proven by Einstein around 1900. It was before electrons - pieces of atoms - were known. Never the less, the names turned out to be reasonable.
When you crowd electrons together, it is like compressing springs. The electrons gain potential energy. If the electrons are more crowded at one end of a wire than the other, the spring forces will push them down the wire.
The important quantity is the potential energy per electron. This is a measure of how crowded they are, and how much energy there is to transport. We call the difference in potential energy per electron between the ends of the wire the potential difference.
We measure potential difference in Volts because Volta was one of the scientists who made important discoveries.
A: Electric current $i(t)$ through a surface is defined as the rate of charge transport through that surface. The rate of charge transport through the surface is not the same as the drift velocity of the electrons through the surface.
For a fixed value of current, the drift velocity of the electrons will vary with the cross sectional area of the conductor. The smaller the area, the greater the drift velocity needs to be in order for the same rate of charge  transport (current) through the surface.
A mechanical analog is the continuity equation for incompressible fluid flow. The product of the velocity of the flow (analogous to charge drift velocity) and the cross sectional area equals the volume flow rate (analogous to current). For a fixed flow rate, the smaller the cross sectional area the greater the flow velocity.
Hope this helps.
A: Consider a circuit set up with a switch and a resistor. Let us think of this resistor to be associated with a lamp. That is, the heat caused by charges traveling through the resistor is what provides light.
The speed of electricity is almost instantaneous. That is, if you are talking about the time it takes between turning on a switch for a lamp and the lamp turning on.
This is because turning on the switch causes a potential difference, creating an electric field which applies nigh instantaneously to ALL charges in the conductor (a very simplified picture). Thus, the charges immediately next to the resistor in the circuit will have a force exerted on them due to the potential difference which starts the flow of charge immediately into the resistor. So, the lamp will instantly turn on. And, this flow will not stop until you turn the switch off.
The drift velocity is how many meters/seconds a charge travels as a result of this electric field.
Crucially, because 1) there are charges EVERYWHERE in the conductor, 2) a potential difference caused by turning on the switch creates an electric field near instantaneously that applies to ALL charges in the conductor, that the speed of electricity (the speed that it takes between turning on the switch and the light turning on) is near instantaneous and is very different from drift velocity.
A: Some accounts of this are pretty misleading, and I'll try to tell it straight though I might fail at that.
First off, electricity is electrons moving. If you have a DC current with a battery, 1 ampere means there are 6 quintillion electrons per second leaving one battery terminal and 6 quintillion electrons per second entering another battery terminal, against resistance. That's what does the work.
Now here are two stories about how it happens. I think the first story is less true, but I can't say it's completely false. Imagine there were no electric fields at all, but electrons could travel freely through copper wire. When you have an electric circuit with a battery in it, there are a whole lot of electrons at one terminal, and electrons that enter the other terminal get involved in a chemical reaction so they don't get out again. So electrons would diffuse through the metal. More of them leave one terminal than come back, simply because there are more of them there. More of them enter the other terminal than come back, because there are fewer available to leave. Diffusion alone would be enough to give us an electric current. As long as electrons travel very fast in random directions, on average they will travel more from high concentrations to low concentrations. That would be enough.
However, we know that there are electric and magnetic forces.
Now I will explain magnetism. I warn you that some people believe my explanation is wrong. So check it for yourself, or be cautious. For historical reasons, we knew there were magnets, and we thought magnetism was something fundamental. But if you look carefully at the equations, there is no magnetic force unless the source charge is moving. And magnetism has no effect on a target charge unless the target charge is also moving. Magnetic force depends entirely on the frame you choose as an observer. It is a fudge factor, because the electric field equations do not otherwise account for frames and are only correct in some frames.
If you're with me that far, then it turns out that electrons that move at relativistic speeds in wires, have some consequences. An electron's electric field is stronger in front of it than behind it. So it is attracted MORE by the positive charges ahead of it than by the positive charges behind it. But it is not affected more by negative charges ahead of it, which are traveling at the same velocity. The very fact that there are electrons traveling through a wire, creates a force to speed them up. So when a circuit is first formed, only a little bit of current travels through it, and the amount increases until it reaches an equilibrium with the things that tend to slow it. And if the battery stops providing voltage, the fast-moving electrons don't stop instantly because their own velocity gives them a decaying force to keep them going awhile. This is called "hysteresis".
There are various other effects of relativistic electron velocity, that explain various properties of electricity.
Well, but there's the claim that electrons in fact don't travel at close to lightspeed. The fields travel that fast but the electrons don't. You can calculate the average speed of electrons in copper wire, and it's very very slow. This might be misleading, though. First off, copper has 29 electrons and only one of them can move. And then maybe the ones that can move are in fact moving at relativistic speeds, but mostly in random directions. When you get an electric current, a small extra fraction of them travel in the direction of the current. Not very many, only a quintillion or so per ampere, but that's enough to give you the observed current. All the rest of the motion averages out. So on average they move very slow, but the ones that make the current could be going fast. They're just a small fraction of the whole.
Well, but I've seen the claim that the average random movement of electrons apart from electric current is fast but not nearly that fast. Maybe I'm wrong.
Maybe all the simple answers are too simple, and the complicated answers are not easy to follow.
A: As to whether it effects the speed of current:
I = e/t and I = nAve
thus, 1 = nAvt
v and t clearly have an inverse relationship i.e. the faster the drift velocity the faster the current (as the less the time period).
A: Electrons don't really "drift" through metal (as current does not really "flow"), they vibrate, with that vibration increasing with things like (but not limited to) temperature.
They may shuffle along atoms a little bit, but not in the sense of flowing or drifting of water.
Electricity does not have a varying "speed" or "velocity", it's more like a resultant output effect for a given set of input effects. Similarly, electrons don't really "propagate" through a circuit.
One Professor of physics suggested to me that there might even be only one electron in the entire universe. The theories you are taught at school are limited to just getting you through that level for a limited range of purposes.
