The number of pieces of a lens doesn't have anything to do with the optical quality. Usually they use 1 piece lenses inside the capsular bag (the bag where your natural lens was that they try to put the replacement lens in), or a 3 piece outside the capsular bag (if the bag tears when removing the natural lens for instance).
The Tecnis lenses tend to be high quality, though typically they'd use a 1 piece IOL for placement inside the capsular bag. I don't know why they are using a 3 piece in your case, if they think there is some problem with the capsular bag which means it might be placed outside the bag perhaps.
They measure the power of a lens, how strong it is, in units of diopters, which is the "D" that is in the IOL power or is used for glasses or contacts.
Your natural eye acts like a lens, but it actually has 2 major parts you might think of as seperate lenses. Lets call lens "A" the actual natural lens in your eye which has a cataract, and lens "B" the rest of the eye. If someone is nearsighted like they are, it means those lenses aren't the right power, and someone gets a prescription for glasses or contacts for a third lens, "C". Since you have a cataract, they need to replace lens "A" with an IOL.
If you had perfect vision now and didn't need glasses, that would have meant lenses A and B were the right power (that you didn't need the third lens, C). So if they are going to replace A, with a new lens, they would want the IOL to have the same power as A. The typical power of a natural lens is about 22D (D=diopters), but each person varies. You can't determine the power of A from the power of your glasses C. If someone has good vision without glasses, their prescription C is 0, but their lens power A is still something like 22D or so.
If you wear glasses now, then as long as they are replacing lens A, rather than giving you the same power as A, they can change its power to give you good vision without glasses. Its sort of like combing the power of A and C into the power of the IOL. The problem is determing what your actual natural lens power is, determining how much of the eye's focusing power is in A and how much is in B. They can't do that with a regular eye test. The way they do that is by taking different measurements of the size of the inside of your eye. They compare it to the measurements of people who had surgery before, and the lens power that worked for them.Your eye measurements won't be exactly the same as anyone elses, but using statistics they can take a good guess about what power replacement lens would work for you.
So they don't use your current glasses prescription for anything. The week of surgery they will give you a pre-operative exam some day before the operation and will take measurements of the inside of your eye to try to calculate the right lens power. The question for you is what distance you want the lens to focus at. When people are talking about getting a lens set for say "-1D", they mean to focus where a slightly nearsighted person would see, which is 1 meter. The actual IOL power is not -1D (it might be perhaps 21D), its merely an indication of how nearsighted you want that eye to be.
If you don't mind wearing glasses, then there is no wrong choice since you can always wear them afterwards. In general terms there are three general distance ranges: distance, intermediate and near, and the IOL is only good for one of them, and maybe a little bit in another. If you get an IOL set for distance, then intermediate will be a bit blurry and you likely won't be able to see much of anything up close. If you have it set for intermediate, then distance will be blurry but you may see some, and near will be blurry. If you get it set for near, then you may have some blurry intermediate, but likely not see much at distance. When I say "blurry", how much depends on the person, and the exact distance. You can always wear glasses for the other distances.
You can set both eyes to focus at different distances, but they can't be *too* different or your brain has trouble combining the two.
Unfortunately the closer in you set an eye, the smaller the range of good vision. e.g. someone set to have good distance vision might see well from 6 feet to infinitiy, but someone who gets an IOL set to focus well at say 10" might only get 1.5" of good vision. That is a result of the math (which I know many people don't like to deal with). The point of best focus is distance_in_centimeters = - (100/ lens_power_in_diopters)
So getting set to focus at -0.5D would be -(100 / -0.5) = 200 cm, or getting to focus at -2D would be -(100/ -2) = 50cm. The reverse calculation to find the diopters for a particular distance is then lens_power = - (100 / distance_in_cm).
You might have a range of good vision of perhaps 0.5D from the best focal distance (however it depends on the IOL and the person, it might be more like 0.75D for some) . So a lens focused at -2D would give some good vision at -2.5D (or -1.5D going the other way, though there is usually less of a range of vision going outwards than going inwards).