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Disclaimer: The information contained herein is for reference only. Always consult qualified technician and take all necessary precautions when working with electricity. Author and Zilvia not responsible for content or misuse thereof.
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Background: As the title implies, this is only a brief guide of all the information an average person would need to know before working on a vehicle's electronics. This information is available everywhere, but I have rewritten it here in my own words for copyright purposes. I hereby release all copyrights under fair use to Zilvia and all members thereof.

Most of the information contained herein could be found in a high school or college physics textbook. More in-depth information can be found in introductory electronics textbooks. If you need to find more detailed information, simply plug any keywords into Google and you should be able to find the relevant information.

Corrections and improvements are always welcome.
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Electrons and charge carriers

Simply put, atoms are comprised of:

* protons - positively charged particles
* nucleus - central portion of the atom
* electrons - negatively charged particles

For reference, an atom is a football stadium. The nucleus is a sphere on the fifty yard line with a radius of about 5 yards. The proton is a basketball contained inside the nucleus. The electron is a golf ball in the bleachers.

This gives you a rough idea of the size. Electrons are the only thing that move because they are smaller and further away from the nucleus.

If the atom has more electrons than protons, the atom is considered to be negatively charged. Conversely, if the atom has more protons than electrons, it is positively charged. If electrons = protons, the atom = neutral.

A positively charged atom, i.e. electron deficient atom, will want to suck electrons in to get to neutral. When an electron moves, this is a current.

Hence, electrical engineering is the art of moving electrons to do useful work.
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(+) and (-) Convention
In the US, a (+) sign signifies an electron deficiency, i.e. a positive charge. A (-) sign signifies an electron excess, i.e. a negative charge. The US convention is that current flows from (+) to (-). A complete circuit flows from (+) and ends at (-), battery terminal or chassis ground.

In other parts of the world, the convention is that current flows from (-) to (+) because this is the path of electron travel.

Keep this in mind as you work on non-US parts. You do not want current to run backward. If in doubt, consult the manufacturer or a qualified electrician.
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Ohm's Law
V = I X R

V : Voltage, measured in Volts
I : Current, measured in Amperes
R : Resistance, measured in Ohms

Just keep in mind that ALL vehicle electronics should be 12V. Check all your components before using.

Each component, including wire, has a resistance. Given 12V and the resistance, you can determine the current draw in Amps.

Note that when you wire components together in series or parallel, the total resistance of the circuit goes up, and current draw will be higher than each component by itself. Search Google to see how to calculate resistance for series or parallel circuits.
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Wire Gauge
If the wire is too thin for the current, there will be excess heat and you risk a fire.

You need to calculate the thinnest size to accomodate your maximum current draw. Remember that the longer the wire, the higher the resistance. It's better to error on the size of lower gauge (thicker wire).

http://en.wikipedia.org/wiki/American_wire_gauge
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Relays
Relays are useful when you need to control a large current, but don't want to draw the current directly through the control source.

For example, the ECU needs to control 30A fans, but you don't want 30A flowing through the ECU.

Instead, the ECU sends a small current to trigger the relay. The relay then draws and sends 30A to the fans. The ECU doesn't need to handle 30A.

http://en.wikipedia.org/wiki/Relay
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Proper Grounding
95% of electrical failures are due to improper ground. You need to complete the circuit by connecting (+) to the component, then to (-), which is chassis ground. Electrons are returned to the battery's (-) terminal.

Always check to make sure your grounds are hooked up correctly.
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Fuses
Fuses are essential. They break when current draw exceeds the limit. This prevents the circuit from drawing too much current and overheat.

Always use fuses with the proper capacity. Never replace a broken fuse with one that has a higher rating. Chances are you have a bad ground somewhere. The last thing you want is draw more current to the damaged circuit.
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Diodes
A diode is like a one-way switch; it lets current flows one way but not the other.

This is appropriate in situations where you don't want current to backwash through 2 circuits.

For example, if you wire a warning light to two different sensors, when one sensor triggers the light, current will flow through the light and to the other sensor, possibly damaging the other sensor. You need to put a diode between each sensor and the light. This way, current can only flow from the sensor to the light, and no current can backwash to the sensor.

http://en.wikipedia.org/wiki/Diode
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Design Considerations
When designing your own circuits, you need to give a lot of thoughts into it.

What are you trying to do? Are you competent with wiring? Do you know how to solder? Do you know how to insulate properly? Did you use the right gauge? Did you use enough grounds? Will the circuit work properly?

It takes a little experience to design a proper circuit with fuses, relays, switches, diodes, and grounds with the proper wire gauge, but if you're willing to do a little more research it will work out for you.

Make sure all your circuits end with proper groundings. You never want dangling components. Plus always use relays when appropriate. Never draw more current than the circuit can handle.

For example, a lot of people piggyback electric fans on the circuit for the fuel pump. Since the fuel pump circuit is only rated for the fuel pump, adding fans that draw a lot of current will most likely trip the fuse. Then they replace with a higher rated fuse. This makes the circuit draw much more current than it's designed to handle = heat = fire = dead idiot.

The proper way is to use a keyed ignition source to trip a relay that draws current from the battery to the fan. Use proper wire gauge for the fans. This is much much safer.

Another common mistake is bad insulation and connection. There is no excuse for this. Take the time to learn how to connect wires correctly.

Also, always document your wiring for your own sake. If you do a sloppy job, it gets worse if you forget what you did later on. Always label your connections and draw out the wiring diagram to make sure you route power correctly.

Make sure you get a clean connection to ground. Sand lightly and use an unpainted surface if possible.
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If anybody has any specific questions/comments/suggestions, please feel free to PM me directly.