Pickup

 A pickup is the starting point of an electric guitar's signal.
Very simple structure : magnets with a very long coil wound around transform the vibrations of strings into an electric signal. (We are talking about "magnetic" pickups, not "piezo" pickups.)
Steel strings vibrating in magnetic fields induce a current in the coil.
Many variations like thickness, types, and turns of coils and the magnetic forces are possible.
Selector

 A signal generated from a pickup goes to the selector.
In 1 Volume/2 Tone or 1 Volume/1 Tone types, the pickups' hot wires go to the selector first. In 2 Volume/2 Tone with 3-way toggle type, the pickups' hot wires go to volume pots first and then to the selector. But the basic role of making combinations of pickups is the same in either type.
The quality and stability of electrical contact and the feel of operating are very important for a good transmission of electric signal.
Volume Potentiometer

 A potentiometer is a variable resistor.
A volume potentiometer controls the amount of signal going to the output by changing the value of resistance.
Usually 250K, 500K, 1M Ω(ohm : the standard unit of resistance) pots are used for electric guitars. Higher value of the pot makes the tone brighter.
As a general rule, 250K is used for single coil pickups and 500K for humbuckers, which results in tone compensation for the bright single coil pickups and the mellow humbuckers.

Capacitor

 A capacitor is a passive electronic component that stores energy in a form of an electrostatic field. It consists of two conducting plates separated by an insulating material called "dielectric".
DC(Direct Current) is cut by a capacitor, but AC(Alternating Current) flows through it. Also a capacitor's behavior is frequency dependent - AC with higher frequency flows better through it. This characteristic is used for cutting treble range of an electric guitar. You can see a capacitor connected parallel to a guitar circuit with one end of it to the lug(s) of tone pot and the other to the ground. Tone pot controls the amount of current goes to capacitor.
Generally, 0.022µF(*) is used for humbucker pickups and 0.047µF for single coil pickups.
Sometimes 0.1µF was used for vintage single coil pickups. The higher capacitance value, the more treble range can cut from the original guitar signal.
Even at the same capacitance value, sound quality differs by manufacturers or material.
Also the quality of capacitor determines the sound quality even at the maximum tone level(10) because some parts of signal would still penetrate through the maximum resistance value(250KΩ or 500KΩ) to the ground.
 * 1 µF(micro Farad) = 10-6 F = 0.000001 F(Farad : the standard unit of capacitance)
Tone Potentiometer

 A tone potentiometer controls the amount of signal going to the capacitor connected parallel to the circuit.
Normally the same type with the volume pot is used, but sometimes linear taper pot is used for the tone control.
At tone level 0, the capacitor's effect gets to the maximum resulting in the maximum cut of treble up to the capacitance value.
Contrarily, most of the signal goes to the output with minimum treble loss at tone level 10.
But the capacitor still affects the tone even at tone level 10 as described above. For example, Fender® is using "No Load" pot for one of the tones on American Series®, which excludes the capacitor from circuit at tone level 10. This means they admit the effect of capacitor even at tone level 10.
Output Jack

 The signal generated from pickup travels through selector, volume, tone, capacitor, and then arrives at the output jack finally.
The quality of jack material, shape, and stability decides the sound quality. Now the signal from this output jack starts a journey to an amp via cables and effectors.
 The basic role of all the cables and effectors is to transmit this signal to an amp without loss of quality or its charms even after the sound processing.
Schematic
This is a schematic of an electric guitar. The selector is excluded for simplicity.
The circle with +, -, ~ marks means the source of AC. In electric guitar, the pickup is the source of AC.
The square with round corners indicates impedance of pickup. Pickup impedance is determined by inductance and resistance of the very long coil wound around the pickup magnets. We will talk about impedance on 2. Impedance in a bit more datail.
Here now, just note that this pickup impedance is around 20 ~ 40 K Ω, which is high impedance compared to the line level impedance 600 Ω(= 0.6 K Ω).
"High impedance" signal means that this signal is very susceptible to the structure and length of cables and easy to lose treble and punch by the cable capacitance. Also this signal is to be easily interfered with ground noise and external electromagnetic waves.
The quality and arrangement of effectors and cables play an important role in a good transmission of the high impedance signal to an amp.

A signal from pickup goes to the volume pot and then to the output jack.
Tone pot and capacitor are connected parallel to the circuit and controls the tone. Tone pot determines the amount of signal "leaking" to the capacitor, and the capacitor lets high frequancy range flow to the ground by its capacitance value.