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The battery and the alternator are 2 of the most important maintenance components for the engine. The battery produces the electricity which will start and run the engine, while the alternator produces the electricity which will recharge the battery¹.

Battery

A battery uses chemical reactions to produce electricity². Like any chemical reaction, certain chemicals ("rectants") are converted into other chemicals ("products"). Should the products contain less energy than the reactants, then the extra energy is released into the environment (electrical circuit). Should the products contain more energy than the reactants, then the missing energy is absorbed from the environement.
Whenever the battery is providing electricity, it is releasing energy of its own through the conversion of reactants into products.
Whenever the battery is getting recharged, it is absorbing energy (provided by the alternator) in order to force the conversion of products back into reactants.
Theoretically, this cycle of battery use and recharge can be repeated infinately, however the process is not 100% efficient. Some of the products and reactants are lost over time³, the battery eventually cannot produce enough electricity anymore and you will experience the symptoms of a dying battery.
Anything which requires electrical energy will get that energy from the battery, therefore a dying battery will manifest itself through many of these devices malfunctioning. Such devices include but are not limited to:
->starter (engine will turn over slowly or not at all)
->spark plugs (engine will not start, will misfire or stall)
->headlights (dim or out)
->signal lights (blink slowly or not at all)
->heater fan (blows less or not at all)
->wiper motor (wipe slowly or not at all)
->radio (weak or no signal, volume low or off)
An alternator in good condition can produce enough electricity to keep the engine and accessories functioning despite a dying battery, however a dying battery may not be able to hold enough of a charge to start the engine without a boost.

Alternator

The alternator uses a belt to siphon engine power to produce electricity. This process is not unlike the process used in power generation plants in that a powerful magnet is repeatedly turned close to electrical wires⁴. Since an alternator has moving parts, these are subject to wear and tear, making the alternator a "consumable" needing replacement every few years. As much as the battery runs the engine, the alternator keeps the battery fully charged while the engine is operating. A failing alternator will not fully recharge the battery, the battery will drain itself of power rapidly (approximately 20 minutes of driving) and you will experience the symptoms of a dying battery until the engine eventually stalls. Diagnosing a failing alternator can be challenging, as a failing alternator can manifest itself in different ways. Typical tell-tale signs of a failing alternator is the engine stalling and being unable to start by itself after stalling; the engine will start perfectly fine after the battery gets a boost and will be driveable but the engine will eventually stall after some time (approximately 20 minutes of driving). An alternator overcharging the battery has been known to cause weird electrical gremlins including blinking dashlights and engine misfire.

Since the alternator uses a belt to siphon engine power in order to produce electricity, changing the diameter of the pulleys used to drive the alternator will reduce the drag on the engine. The alternator will be turning a little slower ("underdrive"), but manufacturers usually design the alternator with up to 25% reserve, meaning you can slow down the alternator by 15% to 20% without affecting the engine or accessories at all unless you are running more than 600W in accessories (big stereo).

References

1. How Stuff Works > Alternators
2. Wikipedia > Lead-Acid Battery
3. Wikipedia > Car Battery
4. Wikipedia > Alternator