An electrical circuit requires a source of electricity, something to conduct it, and something upon which it can act (resistance).
Your source in the aircraft is the aircraft battery, and the alternator/generator. The battery is simple; it has a positive terminal and a negative terminal. Electricity flows from the negative terminal to the positive (contrary to popular belief). The difference in electrical potential between the two is voltage. Batteries are typically either 12 volt, or 24 volt. A volt is nothing more than a measure of the force trying to get from one terminal to the other, when the values of the two are compared.
As another poster noted, volts can be compared to pressure in a hose. Volts is the force pushing the water through the hose.
Amps or amperes, is a value which represents the current or flow of electricity. Think of it as the amount of water flowing through the hose, and think of it in terms of how it could hurt you or drive a motor. More current or more flow (more amps), the more it can hurt. High volts with few amps, not so bad, but high volts with lots of amps; that could hurt.
Watts could be thinking of what the electricity is doing or capable of doing. Think of a 60 watt lightbulb. More watts, more capability.
Batteries are rated in ampere/hours. A strong amp/hour battery is capable of sustaining more current for a longer period of time than a smaller amp/hour battery. Batteries are also rated in terms of cold cranking amps, which is what they can put out for a shorter period of time, and it has application with putting a heavy load on the battery, such as during the engine start.
All alternators and all generators put out alternating current. That means that electricity changes direction in a wire repeatedly. Generators use devices inside or outside the generator which filter the electricity to allow it to go just one way through a wire, just like the electricity that a battery puts out. When the electricity travels one direction through a wire, it's called "DC," or Direct Current. Sometimes you'll see electrical value written as "VDC," meaning volts of direct current.
Conversely, an alternator puts out AC current, which keeps switching direction in the wiring. In most light aircraft, this isn't useful for the simple electrical system, and small devices called diodes are used as electrical filters, allowing only DC to the aircraft sytem. Large aircraft use AC electricity (Alternating Current: same as you get out of the electrical outlet in your wall) for most components, with some parts of the system converted to DC electricity by "rectifying," which is a lot like straining the electricity to make it go just one way in the wire.
AC electricity is a little harder to control, especially when several sources of electricity are used at the same time (multiple alternators, for example). As long as DC electricity has the same voltage, it can all be sent to the same wire or same devices. DC is simpler to use, and is the most common in light airplanes. Aircraft systems are usually either 14 volt, or 28 volt. The batteries are 12 volt or 24 volt, and the slightly higher alternator/generator output is used to help charge the battery.
Think of the battery as an electrical piggy bank. It stores electricity. It also helps act as an electrical shock absorber. If a sudden change in electricity occurs in the airplane, the battery helps absorb that electrical spike, and creates more even flow through the aircraft system.
Ammeters show the draw on the battery; a positive value or close to neutral value shows that the alternator is doing it's job. A negative value shows that the battery is being discharged, and isn't being charged by the alternator.
Voltmeters tell you what the highest voltage is in the aircraft electrical system. A value lower than the nominal battery voltage tells you that the battery isn't fully charged, or is being drawn down. A voltage higher than battery voltage tells you that more electricity is available in the system than the battery puts out, which tells you that the electrical system is working, and implies that the battery is being charged.
Too high a charing rate isn't good; it tends to make batteries hot. This can be seen sometimes if a battery is really drawn down, such as if one really cranked on the starter, trying to get the engine to turn over. If the battery is depleted by the time the engine starts and the electrical system is engaged (alternator turned on), the sudden rush of electricity to the battery as it's recharged can make it hot. Some types of aircraft batteries, especially Nickel-Cadmium (NiCad) can experience a "thermal runaway" in which they keep getting hotter and hotter and have an internal breakdown which can melt the battery. In most light airplanes, these types of batteries are not usually used. Instead, the same type of battery that you find in your car, a lead acid battery, is used.
Your aircraft usually uses what's called a "negative ground" system. Because electricity flows through a conductor (wire, etc), with end connected to the negative terminal of the battery and one end connected through the positive terminal, aircraft use a simple method of flowing electricity. The frame, fuselage, or certain parts of the aircraft are connected to the negative terminal on the battery. This is called the "ground," or sometimes the "earth." The electrical components of the aircraft are hooked to the positive terminal through wires, switches, circuit breakers, etc. The other side of the electrical components is hooked to the frame. This reduces the wiring in some cases, and simplifies the way the electrical system is set up. Your car uses this method, too.
Electrical systems in the aircraft are hooked to metal bars that feed a number of wires. Each of these bars or groups of bars is called a "bus." Light aircraft often only have one or two busses, whereas large or complex aircraft may have dozens. These are basically like outlets in your house. Electricity goes to the bus, and then various components get their electricity off the bus. You'll see this term often used in aircraft flight manuals in the systems descriptions.
Circuit breakers exist in circuits (a circuit, remember,is the electrical source, the wires that conduct it, and the motor, light, or other resistor that's in the circuit) to protect the wiring in a circuit. These breakers are simple switches that usually open automatically if too much electricity is forced through the wiring. The circuit breakers are safety devices that get warm or trip magnetically, and open the circuit to stop the flow of electricity.
Understanding your electrical system is important: it lets you know how to recognize what's wrong, what's right, and what to do about it, and how to operate it safely and efficiently. In most cases, there's little to do as a pilot, but if system components fail (such as a generator failure), then knowing how to reduce the electrical load on the battery by knowing which components use the most electricity (lights transponder, etc), then you can make your electricity last longer. If you know that your flaps are electrical, you might consider not landing with the flaps, in an electrical failure situation, because if you need to go around, you might not have enough electricity to retract the flaps. If you have a retractable gear airplane, you might know that a failure of the gear to extend or retract might be a hot solenoid, and shutting off the electrical for ten minutes might fix the problem.
Knowing your system and how it works, unique to the aircraft you're flying, helps you know how to handle problems, and how to handle it when it's functioning properly.
Most light aircraft using piston engines use magnetos. These keep producing their own electricity and will keep the engine running even if the battery goes dead or gets shut off, and even if there's no power from the alternator/generator. These are simple devices which produce electricity for the spark plugs, and then distribute that electricity to the spark plugs. They are distributors just like you have in your car, but differ in that they can also produce the electricity; your airplane will have two each supplying one of two spark plugs in each cylinder, to help ensure that your engine will keep firing away even in the event of an electrical failure, or a failure of a single magneto.
