WARNING TO ENGINEERS: "Laymens terms" description ahead. Do not cringe. 
An amplifier is an electrical or electronic device that uses a small voltage to control a larger voltage. Small voltages (like the signal off a CD) are used to control a larger voltage in the head unit to create the line level signals. This is called a pre-amp. These larger voltages are then used to control an even larger one to create the speaker level signals. This is called a power amp. Both amplifiers work in pretty much the same way, and both can clip.
For a very gross physical way of visualizing this, think of a garden hose with a nozzle on it. You and the nozzle work together as an amplifier when you control how much water can come out of the hose. You are not creating the water, it has to come from somewhere (like your house, which is car audio terms would be the alternator and battery) but you can control how much is squirting your lawn at all times by twisting the nozzle or opening and closing the valve.
Just like you and the nozzle, an amplifier does not create anything. It controls or regulates a larger voltage. Other circuitry inside the equipment (called the power supply) makes sure the large voltages are available. Sometimes these voltages come straight from the main power source (like the battery) and sometimes they are generated in the power supply through a variety of electrcital and electronic means to be much higher than what the battery can provide. The ratings of amplifiers are determined by the maximum output voltage it can make available along with the maximum current flow that can safely pass through the amplifier circuit.
Clipping is what happens when the amplifier tries to put out voltages higher than what is available from the power supply. Once the input (the small voltage) demands 100% of the available supply voltage, the output cannot go higher. It is possible and very common, however, for the input to greatly exceed the 100% mark. When this happens, the amplifier circuit says "OK, you want more, I'll open up my nozzle wider." But if you've already hit the max your power supply can give you, or if the valve is already all the way open the flow cannot go higher no matter how hard you twist it. You can only get the maximum the supply has available. This in effect means the output during the clip remains constant at this maximum value. Constant voltage is called DC (direct current.)
DC is bad in the sound world for many reasons, but it is especially bad for loudspeakers. Basically it means the speaker stops moving during the DC part of the signal and all the power being supplied to it generates heat. A speaker makes sound by moving and DC sounds bad, and a speaker cools itself by moving, so DC keeps it from being cooled. The ultimate result is what Rob at Forbidden calls "SSS" (smelly speaker syndrome.) Severe clipping can also cause the output amplifier circuits or the power supply to become smelly.
The "gain" or more accurately the input sensitivity control on a power amplifier is only there because there is no universal pre-amplifier level standard in car audio. It is unfortunate in many ways because it enables novice users to easily destroy their gear. It is also a very powerful control because it allows knowledgable users a lot of flexibility in their equipment choices. Regardless, what it does it set the amplifier sensitivity to the maximum expected "small voltage" input signal. If the gain is set so that the amplifier will reach maximum output at the same time that the input signal reaches maximum, this is called "unity" and is in general the ideal setting for the control, but any value less than this setting is safe to use. If, however, there is a level mis-match so that the input signal can exceed the maximum expected value (for example, your input gain is set to 2 volts but the HU can actually generate 4 volts) the amplifier will clip before the input signal reaches 100%. This is bad.
When you turn the gain setting "up" you are actually setting the amplifier to expect a lower input voltage. If the setting is off from what the input voltage can actually achieve, clipping is the result.
I hope that helped. If not, here's some links. Read Here, Here, or Here.
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