pickups are so named because they "pickup" the "sound" of the guitar strings.  I say "sound" in quotes because, unlike with a microphone that listens to the vibration of the air, electric guitar pickups get their signal from the motion of the guitar strings.

This is accomplished by "electromagnetic induction,"  the same phenomenon as most power plants run on.  The concept is simply: physical movement of metal in the presence of a magnetic coil of wire will produce electric current in that wire.  At your local power plant, either fossil fuels or nuclear generated steam turns turbines.  This moves pieces of metal through the coils.

Similarly, movement of the metal guitar strings over the magnetic field produced by the pickups will produce a tiny electrical signal that pulses at the same rate as your string vibrates back and forth.  This is a very small signal, but that's why we plug into an amplifier (so named because that's its job!).

The entire output can be cut by the volume knob.  This is because a variable resistor sits between the guitar pickup(s) and the output to the amplifier.  A variable resistor is more commonly known as a potentiometer because is alters the electrical potential in a portion of a circuit.

When the knob is turned to "10," the entire signal bypasses the resistor, so there is no resistance.  However, if the knob is turned lower, the amount of resistance increases.  This means that some of the signal will be blocked by the resistor.  If the knob is turned all the way down on most guitars, all of the signal will be prevented from reaching the output.

Note that guitars with humbucker pickups usually have higher value potentiometers (500k Ohms) than those with only single coil pickups (250k Ohms).  This is because humbuckers produce a larger signal/higher output thanks to the way their two coils are connected.

Just like volume knobs, your tone knob is a potentiometer that cuts part of the signal.  In this case, the tone knob works by cutting only the higher frequencies in the signal.  This is accomplished by placing the potentiometer on the "side" of the circuit rather than in the middle.  Part of the signal is run to the electrical ground.

The reason the high frequencies are cut first is that they have more energy (just like x-rays).  If you think about it, what sound comes through a wall?  The bass!  All the treble end of the spectrum are filtered out by the resistance of the wall between you and the source of the sound.  It's the same principle here.

3-way switch -  In a Les Paul or other guitars with two pickups, a three way switch selects between the pickups.  It also can select both pickups at the same time.  This is accomplished by the toggle portion pushing the contacts to the appropriate pickup.  (Note that the pickup selected is actually on the opposite side of the switch than you would think; This is just a quirk of the design.)

5-way switch -  On Stratocasters and other guitars with three pickups, this type of switch selects the individual pickups as well as combinations of the middle pickup with the other two (at least, that's the typical configuration; some guitars --mine, for instance-- sometimes have a few twists in their design).  This is because the "in-between" positions make contact with the pickups in the adjacent positions.

A 5-way switch also has the additional feature of being built around two "poles," so it has two independent circuits on either "side" (I say "side" in quotes because these switches come in a confusing variety of configurations).  These poles allow for additional connections, including to the individual tone knobs as on a Stratocaster.

The lugs on each "side" of the switch are numbered like this:

0 = "common" (all positions will connect to here when selected)
1 = bridge pickup
2 = middle pickup
3 = neck pickup

To see how these are connected in a conventional guitar, see the stock Stratocaster schematic.  Of course, if you're in doubt, you can always run a continuity test across the lugs to figure out which connect to which in each position.

Note: There are other switches out there that have more than two poles.  These are usually clearly marked as such (they also have more than 8 lugs to connect to), so it's kind of obvious.  These allow alternative connections that you can learn about elsewhere on this site.