Unfortunately the audio companies are forced to overly simplify things for marketing purposes, and it makes it difficult to compare the specifications of different pieces of equipment.
Exact power ratings for amplifiers and speakers are irrelevant, and they aren't really useful unless you also have some other pieces of information (more below).
Speaker are commonly given a rating in watts rms, which is essentially the average continuous power you can run into them before you burn out the voice coil or exceed the cone's maximum travel (xmax). The other important number to know is the efficiency of the speaker, which is often measured as the number of decibels (db) it produces with 1w of input. With a high efficiency speaker you need less input power to get the same sound level than you do with a low efficiency speaker. Because of this one "45w" speaker may be nowhere near as loud as another "45w" speaker.
Amplifiers are rated in watts as well, but this rating alone doesn't really tell you much. The amp's output varies widely based on the volume level and the content of the music, and you'd be surprised how often the amp is only putting out a couple of watts during normal listening (Google "first watt"). The other important number is the distortion level at the rated power output. As you crank the amp up it will distort more and more, which makes it sound bad and can damage your speakers. The power rating (watts) is based on a given distortion level, for example 100w @ 1% THD. Just to make it confusing, a high quality amp rated at 35w @ 0.01% THD puts out much more clean power than an amp rated 50w @ 1% THD. Luckily amp power rating methods have been somewhat standardized over the last 5-10 years.
Ok, so what does this all mean if you want to upgrade your stereo and you don't want to do hours of research?
1) Don't fixate on the specific watts ratings of speakers or amps. Buy from reputable manufacturers, in general aim to have the rated output of the amp be higher than the rated Wrms of the speaker so you hopefully have enough clean power. For normal listening levels speakers rated around 30 Wrms or higher should be fine.
2) Go with 4 ohm speakers for mids and tweeters in car audio, this is the standard.
3) Keep the signal to noise level high, this means adjusting your amplifier's gain so that you get the most voltage out of your head unit as possible. In other words, set the amplifier's gain low so that your "as loud as I ever listen to" level is at ~90 of the max volume on the head unit.
4) Don't spend a lot of money on interconnects and speaker cables. You of course want cables with good shielding, but you'll get much better results by spending the extra money on the speakers and not the wires.
5) Consider adding some sound deadening, this lead to major improvements in the bass and mid bass.
- Matt