You're in it knee deep now buddy........
Combining the description of WATTS being the amount of work being done, with a side of the well described efficiency and a splash of ohms law stating a 4 ohm speaker will require twice of an 8 ohm speaker you should be ready to sit down and take a deep breath.........
All of the referenced replies are correct to their respective areas but trying to cross reference them will leave your head spinning.
Watts come from the amplifier. The more you have, the better. Did you ever meet anyone who wished they bought a slower car or smaller TV. Quality amplifiers are rated at a specific load over a specified frequency bandwidth, say 50 wpc @ 8 ohms both channels driven from 20-20000hz. Better quality amplifiers will also specify a 4 ohm power rating and some monster amps may list a 2 ohm rating. Usually as the ohms go down, the power goes up, so your 50wpc @ 8 ohms may be 70wpc @ 4 ohms. That description is typical and does not follow Ohm's law because of limitations in the amplifiers design. Some of the best amps made will double the power into 4 ohms and very few will double that into 2 ohms but by then your ears are bleeding and your bank account is sucking wind. For now 50wpc @ 8ohms is a good starting point.
Speakers are a whole different animal. They come in all shapes and sizes and they all sound different. Which ones are right for you depends on several things. How much space you have for speakers, how much space you are trying to fill with sound, how loud you want it to be and what you listen to all play a part. A simple rule of thumb is the size of the speaker is relative to the size of the space you want to fill with sound. Headphones are small, just like our heads relatively speaking. A speaker that can fill the bathroom with sound usually wont cut it in the living room. listen to everything you can and pick the ones you like best because in the end only you have to be satisfied with your choices.
All speakers are rated in ohms, typically 4 or 8 but there are some oddball 6 ohm speakers just to muddy the waters a little more. Make sure your amplifier is rated for 4 ohm speakers since there are many that are not. If you choose to run 4 ohm speakers on a system rated for 8 ohms proceed with caution and watch out for overheating. Not saying it cant be done but in the interest of longevity and overall safety you probably shouldn't.
Speakers are also rated in db as described above. The higher the number, the more efficient the speaker is. That doesn't mean it sounds any better or worse than another speaker just that it turns watts into loud more easily. If you have a pair of 88db speakers and replace them with a pair of 91db speakers they will sound twice as loud but not necessarily better. Once again, let your ears decide.
Bi amping is the process of using separate amplifiers for the woofers and tweeters in a two way system. This is for the "advanced" hobbyist, the jury is still out if it is truly better or just different and more expensive.
Welcome to AK, remember the first order of business is to ENJOY YOURSELF. As long as it sounds good to you, nothing else really matters.