I work in clinical research with transplant patients (kidney, liver, and pancreas) so people are always asking me questions about transplants. So here's some of the answers I've given.
Almost everything in your body can be donated. Organs, of course, but skin, muscles, tendons, bone, even cartilage and collagen can be reused. These are washed and treated in such a way that they usually cannot be rejected from the body. People who aren't eligible for organ donation can often donate the rest of the body. Skin is usually used for burn victims - the cadaveric tissue is laid on top of the burned skin and helps it fight infections, as well as maintain moisture. Tendons and ligaments are used for reconstructive surgery - if you tear a tendon and have it surgically repaired, often it is simply replaced with a cadaveric tendon, since yours won't grow back together very well. Hearts not suitable for donation as a whole can have the valves removed and used for valve replacement surgery. Bone is ground up or cut into specific shapes - if you have a large tooth removed, your dentist may fill the hole in your jaw with bone dust, which will provide a framework for your own bone cells to grow over it.
Not everyone is a candidate for organ donation. Most people who donate organs are younger (under 65), healthy, and usually developed brain death after some sudden trauma such as a car accident. Except for a few isolated cases, everyone who is an organ donor is completely brain dead and is not expected to live more than a week, maybe just a few days. The family is asked to fill out an extensive medical and social history in order to find out if any other factors such as drug use or a history of cancer are present, which may also preclude donating. The restrictions on who can donate tissue are much stricter concerning disease and possible disease exposure, but the age range and cause of death allowable is greater. However, most people who are dying are not completely brain dead, and during the course of their illness may have contracted infections that may render their organs unusable (for instance, pneumonia would mean that the lungs could not be donated.) You may be able to donate tissue, however.
We don't force anyone to donate - but we may press the issue. A lot of people don't really think about what they want to happen to them in the case of brain death - so most areas have specialized, outside healthcare workers (who work for organ procurement organizations, or OPOs) who will meet with the families. Organ donation can be a really touchy subject within families - I've heard of cases where the husband was brain dead, and the wife and his children knew he wanted to donate his organs, but his mother and father didn't want to do it - so the donation was done secretly so as not to upset anyone. If you don't put it down on your driver's license, and your family is thoroughly against organ donation, they won't be pressed about it. But if you put on your license that you agree to be an organ donor, your family will be asked about it, and asked to reconsider if they are against it, since you have basically signed a contract with the state. If your family is really against it or says you changed your wishes, the OPO will respect your family's wishes.
Your organs are probably going to stay in your state or one of the neighboring states. You've probably seen some person on TV flying across the country with a cooler full of organs. However, what usually happens is that the organs are placed within the territory belonging to the organ procurement agency that covers the hospital in which you died. There are about 50 OPOs, but they don't correspond to state lines. There are so many people waiting for organs - especially kidneys - that the need will probably be local. Hearts and lungs are rarer (both because the need is lower and because they are often unusable), and may be sent out of the state.
Your organs may not be in good enough condition to be used anymore. If you've been in a car accident and were wearing your seatbelt, you may have sustained serious organ damage, so even if you're brain dead your organs may not be in good enough condition to be transplanted into someone else. The same goes if you have had some previous disease or which damaged your organs. Especially for your liver or kidneys, a healthy person may have damage without even knowing it, since you don't need full function of your liver or kidneys to be healthy. Those organs, which work perfectly fine in you, might not be in good enough condition to work in someone else, though.
There is a waiting list - and no, it's not ranked by your race, your income, or where you live. Those factors CAN affect how quickly you get an organ, however - certain minorities are genetically very different from Caucasians or African-Americans (who are the largest pools in America) and so may not be able to match with an organ from a person of a different ethnicity. Your income and where you live may affect with what center you are listed - some centers in some cities simply have more deaths, or their OPO covers a larger territory, or people in that area are more likely to donate. This means more organs come to certain centers than others. Until there is a national network, this is simply the way the system works. Sometimes you can be listed with two different centers, or you may receive special priority if you require a rare transplant (such as a combination heart-lung) or if you are a child.
There is no actual numbered list. You aren't ranked on a real numbered list, because each organ has special criteria that must be analyzed to determine which person would match best with it (such as the donor and recipient blood type, antibody reactions, size of the organ, type of disease in the recipient), etc etc etc. No two organs are alike, and no two organs are best for the same person. However, in general, the sicker you are, the more likely you are to receive a particular organ. If you have a rare blood type, you may wait longer - it all depends on whether someone with your blood type becomes an organ donor tomorrow or in two years. I have seen people go on the waiting list and get an organ in two days - and I have seen people who have waited for three years. Most organs have a standardized form which produces a score that ranks how sick you are, and that is what is used by the doctors to determine the priority.
Transplanted organs don't last forever. While many people may keep their transplanted organs for 15 or 20 years, for most people you are pretty lucky to go five years without a rejection episode. Rejection occurs when your body's immune system decides to fight the transplanted organ as if it were an invader - which means that the organ stops working while it is being attacked. Transplant patients stay on immunosuppressive medications for the life of their organ (and sometimes after) - but your immune system can't be completely suppressed or you would get infections every day. So a delicate balance must be struck between preventing rejection, and preventing infections. For some people, they may go the rest of their lives with the same kidney - but many have multiple transplants.
You can donate a kidney or a lobe of your liver to someone. Many studies have shown that you can live quite nicely on one kidney alone - after all, that's what people with transplanted kidneys do! It's an extremely safe procedure to donate a kidney to someone - you have to be of the same blood type, and you have to be healthy, but other than that, most people can donate without problems. For livers, it depends on the amount of disease damage whether the person will need a whole liver or can accept a single lobe. Your liver has three lobes, and if you donate one lobe, it will grow back in about five to ten years, with no ill effects to you in the meantime. (Although you should probably not drink or do drugs.) This is also a very safe procedure.
They don't take out the kidney through the back (or leave you in a tub of ice.) A transplanted kidney is placed in the front near the belly button - a healthy kidney is smaller than a deck of cards, so it doesn't take up much room. Depending on the situation, the surgeons may leave both of the diseased kidneys in place. Livers and other organs have to be placed in the correct position, however. Kidney transplants leave about an 8-inch scar; other organs may be slightly larger, but except for heart and lung transplants you don't get slit up the front and opened like for open-heart surgery. Much of the surgery is done by touch.
Medicare covers most organ transplants. Organ transplants are generally considered life-saving procedures and they are covered. Kidney transplants are considered life-enhancing (because you can still live on dialysis) but they are covered because no matter the cost of immunosuppressant medications ($800-$1000 for the rest of your life) it's still less than the cost of dialysis ($1000 per visit, 3 times a week for the rest of your life.) My center is one of about twenty in the country pursuing islet transplants (islets of Langerhans are the portions of your pancreas which produce insulin, which diabetics lack) and Medicare is supportive of this as well.
There are a lot more people on the waiting list than there are organs for them. This is a really sad fact of working in transplant - many of the people who are waiting for a cadaveric organ will never get any. It's a cruel catch-22 that the longer you are on the list, the better your chances are of getting an organ, but if you get too sick you may no longer be cleared to undergo surgery and you can't get an organ anymore. So a lot of people kind of "fall off" the list because they get SO sick that it isn't safe to put them under general anesthesia and perform surgery anymore. A lot of people die while waiting for a transplant.
The one message I really try to bring to people who are curious about transplant is this: If you donate an organ, you WILL save a life. Probably MULTIPLE lives. You have one heart, two lungs, two kidneys, one liver, and one pancreas, not counting all the tissue in your body. You have seven donatable organs. Most likely not all of them will be in condition to donate, but most healthy people who have brain damage and are brain dead can donate at least four. This means when you donate the organs of a loved one, at least four people's lives will be changed forever - they will be able to go back to work, to get out of bed, to live their lives again, and extend their lifespan by years and years. Transplant patients are truly the nicest people I have ever worked with, and they are forever grateful for the sacrifice made for them - because they know that someone had to go through something terrible in order for them to live their lives again.