An amplifier has two sections.
pre-amp
This bit has all the tone controls (these switch in varying capacitors and inductors to filter out parts of the signal), a phono stage (which boosts the very weaks signal from turntable cartidges, and equalises them to a "flat" signal in terms of bass and treble), the input switches and the volume control which is a variable resistor, that reduces the voltage to a level that equates to the loudness you want. Speaking of loudness that is usually a setting whch introduces both a bass and treble boost.
power-amp
These amplify the voltage that the pre-amp outputs using transitors or rarely these days valves.
They contain big power supplies to enable the increased voltage to be deliverd with the appropriate current. The power of the amplifier is determined by the size of these power supplies.
The drivers ( as sft42 said) are the bit of the speakers that make the sound, often there are two types, the woofers or bass units and the tweeters or treble units.
The current from the power amps enters coils of wire in the drivers, these coils are placed around powerful permanent magnets. The varying current in the coils makes them move when in a magnetic field, they do this and are normally attached to a cone or dome of paper, card or plastic etc. which is what is visible at the front of the driver.
Ultimately all they are about is turning a varying electrical current into movement that closely (as possible) matches the current. It is this movement of air that we hear as sound.
There are other types of drivers, for example electrostatic, which don't uses magnets but use a thin film of conductive plastic, stretched between electrostatically charged plates. Alter the charge on the plates with the output of the amplifier, the film moves, the air moves, sound is made.
I appologise if I've been a bit simpilstic but you can ignore the bits that are too obvious.