I've read in some places that electricity is very slow. Then how can Internet traffic in copper wires be so fast? I think I have a great gap of knowledge about the low-level data transfer structure. Anyways, I searched for it and couldn't find, so?
Electricity in wires is essentially the movement of charge-carriers such as electrons. These move very slowly.
However a change in voltage at one end of a wire causes a corresponding change at the other end of the wire almost instantly.
Think of a very long hose pipe, if the pipe is empty when you turn on the tap, it can take a few seconds for the pipe to fill with water before it starts coming out the other end. If the pipe is filled with water already, turning the tap in will instantly force water out the far end.
Voltage is roughly analogous to pressure. A change in water pressure can be transmitted through a pipe full of water faster than the water moves.
I actually wrote a blog post on this exact question a few years back.
Basically, though the individual electrons move at only a few millimeters a second, the "signal" of their bumping into each other moves much faster than that (a large fraction of the speed of light).
Note how the balls enter the tube very slowly, but the "signal" that a ball has entered (the force propagating down the tube) moves much faster than that. Electrons moving in a wire work in a very similar way.
The speed at which electrons move in a conductor is relatively slow, but information is carried on pairs of conductors or in the air by means of waves of electricity, and these waves travel very fast: at the speed of light in a vacuum or at about 2/3 of that in a coax cable.
If you take a rope, attach one end to something or have a friend hold it, then quickly flip your end of the rope up and down, you will induce a wave in the rope that travels quickly from your end to the other end. The particles of rope don't move very fast, though.
Another example is an ocean wave. A tsunami travels at hundreds of miles an hour, but the water carrying the wave doesn't move that fast.
Electricity itself travels through wires at the speed of light x velocity factor. So a cheap piece of RG-58 coax cable with a 75% velocity factor can transmit radio waves at 75% of the speed of light.
The confusion comes in because we don't use just DC current to transmit information, but rather an alternating current or an alternating current impressed on a DC voltage (modulation) to do this. Each transmission medium used has a maximum frequency at which it will operate. Telegraph using switched DC 5-50wpm (a wire suspended on poles with an earth return), A standard POTS line may pass 4kHz (rural parallel 2 pair cable with shield), a DSL line close to the DSLAM might pass 2MHz (twisted pair from DSLAM to house), a CAT5 Ethernet twisted pair cable 100MHz and so on. Typically, the higher the frequency that the transmission medium can support, the higher the datarate that can be passed through it.
The electricity passes through the cable at the same speed, but the increasingly higher switching rate carries more and more information.
On top of this concept of using electricity to transmit information is the type of modulation used.
Close in, simple square wave serial may work just fine. It comes in many flavors. TTL uses a 0-5 volt shift, RS-232 uses a voltage swing between positive and negative 3-15 Volts. etc. etc.
For POTS phone lines, transmitting a square wave would not work, the characteristics of the line turn it into a sine wave, so various modulation schemes are used like amplitude shift, frequency shift, phase shift or a combination of these used in QAM which can be used to increase the datarate from 150bps to 56kbps.
DSL operates at a much higher frequency than POTS and using a modulation technique called Discrete Multi-tone, operates over multiple carrier frequencies (think radio AM band and receiving all the stations at once, each one carrying a data signal and combining them at the other end) using phase and amplitude shift modulation, reaching a datarate of 1-7Mbps.
This covers local loop, when we get into the internet backbone we start using stuff like Digital Signal (DS) and Optical Carrier (OC) media to move the traffic using PCM (Pulse Code Modulation) techniques over multiple channels. These are the true digital part of the network. Once again the traffic moves at near light speed, but the datarate is dependent on the frequency and modulation technique being used.
Addendum: Electrons move from one atom to another a minute distance. The current flow we perceive as electricity is a near instantaneous cascade of electrons moving from one atom to another over the length of a conductor.