In order to define your needs, you have to set the results you want to obtain from your system. Just answer the questions a Car Stereo Specialty Store would ask you: What kind of music do you listen to? Do you need great amounts of low frequency bass output? Do you plan to install a subwoofer? Do you listen to music at loud volume? Do you own a convertible or an old car with a loud powerful engine?
If you like listening to music at loud volume and plan to install a subwoofer with lots of bass output, or if you own a car which is very noisy on the road, like open air convertibles are, you will need a powerful amplifier. For a basic installation, a 50 W x 2 or a 50 W x 4 quality amplifier will work wonderfully to amplify a factory speaker system or entry level aftermarket coaxial or component speaker system.
For instance a basic audio system could be made up of two 165 mm (6 ½'') coaxials in the front doors, and two 6” x 9” coaxials installed in the back doors or on the rear deck. In this case one 4-channel amplifier would be a perfect choice; one amplifier channel for each speaker. If your budget doesn't allow you to purchase a 4-channel amplifier, then a 2-channel could be a good alternative to power the front speakers by themselves, using the head unit to power the rear speakers. You could also choose to have both the front and the rear speakers powered by the amplifier, although in this configuration you would lose front to back “fader” control.
In order to make sure you generate the “clean” volume or output levels you desire, always choose speakers that are rated to handle the power of your amplifier. In regards to achievable output levels, it is a big benefit to choose speakers with high efficiency: otherwise you may risk pushing your amplifier to its limit to achieve the desired volume level. This will result in reduced sound quality and even distortion that could eventually lead to damaged speakers and/or amplifiers. It is always wiser to choose an amplifier with slightly higher power rating than your speakers are rated at. In this situation, your amplifier will perform with better sound quality, run cooler and thus be a lot more reliable. The amplifier will then have the headroom and dynamics to bring music to life!