Head Unit

The head unit is the tuner, cassette deck, or CD player that sends the signal to the rest of your car audio system. Some head units have amplifiers built in (in which case you must make sure your speakers are efficient enough to play loudly with the relatively small amount of power in most head units - See the Power Up! section.). On a budget? Buy speakers first. Better speakers can make your stock head unit sound really good. You can upgrade it later. And you will want to. One thing to keep in mind: make sure the head unit has pre-amp outputs when you buy it. You'll need them when you're ready to add amplifiers later.
Speakers

Ultimately, the head unit source sends its signal to the speakers. Your speakers determine how your whole system will sound. No equalizer, amplifier, or processor can compensate for poor (or poorly installed) speakers. Even if you're on a budget, you should plan on spending the bulk of your allotted expenses on your speakers. (And if you're really on a budget, plan on a head unit and a set of speakers now, and worry about amps and processors later.)
Subwoofers

Subwoofers are the speakers that deliver the lower frequencies of the audio spectrum. They need to be specially installed, usually in a box designed specifically for them. They demand more power to play at acceptable levels without distortion, which brings us to...
Amplifiers

An amplifier boosts your signal power, resulting in a cleaner sound and more volume. And because more power is a good thing, an amplifier might be the next thing on your list. Be careful, though, because if you are planning on adding several high power amplifiers you may need to upgrade your car's electrical system with upgraded capacitors, battery, and-lastly-alternator.
More about amplifiers

Amplifiers can really turn your system on. With more power you'll achieve a cleaner, more dynamic sound at higher volumes. But installing an amp yourself can be tricky. Be sure to plan your install carefully.

Never mount amps or other components directly to the metal of your car. (That's just asking for noise problems.) Instead, use screws with rubber isolators when you have to mount to metal, or mount the component to a non-conductive board and then mount the board to your car's body. And before you drill holes to mount anything, hook the component up and give it a test run in your chosen location. How smug will you feel after finding that noise problem can be fixed simply by moving your amp to a new location… before you've drilled?

Amps are sensitive to electrical and motor noise, and they can interfere with your radio reception. They should be mounted at least 3 feet away from your head unit.

You can mount an amp under a front seat. This is close to your head unit, so you'll be able to use shorter cables to both the head unit and the speakers, but larger amps won't work here.
Mount an amp on the passenger side firewall; you won't have to remove the seat, but again only a very small amp will fit.
Better yet, mount your amp in your trunk, where it will have plenty of room to breathe, which is important because…

Amps produce a lot of heat. You'll see cooling fins on an amp. They radiate that heat into the surrounding air to help cool the amp. For these fins to operate properly, they need a few inches of air space around them at all times. Also, try to keep them vertical. Amps should not be mounted with the fins facing downward (because heat will radiate back up into the amp).

A tip:
Just because something is metal doesn't mean it's a good ground. Ground your amp directly to areas of heavy chassis metal only, not to a piece of metal that's merely attached to the chassis. Use ground wire run-lengths of 18"-24" maximum.

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