Organ transplants are less a problem in themselves than in two other questions. In the case of live transplants (where the donor is alive at the time of transplantation), the central question is whether the donor is allowed to endanger him or herself, and also to suffer the injury involved. It is generally prohibited to injure oneself, except for a valid reason (so that, for example, life-saving surgery is clearly ok, but self-mutilation is not only a bad idea, it is prohibited by Jewish law). This has to do with remembering that Judaism does not endorse the view that we are in complete ownership of our bodies; they were entrusted to us by God, with certain parameters around their use, and wanton self-injury violates those parameters.
That is why, years ago, R. Moses Feinstein ob"m was opposed to kidney donations. Since, at the time, the technique had not yet been perfected, and there was some danger to the donor (along with some uncertainty as to how well the transplant would take), he felt that it was taking on an unacceptable risk. With advancements in medicine, it has become more widely accepted. Lesser donations, such as blood, platelets, and bone marrow, are generally seen as meritorious, let alone acceptable.
When it comes to donation after death, another issue arises. Current medicine requires that the organs be removed from the donor body either before or within short minutes after the cessation of circulation, because the organs began to deteriorate so quickly after that. In the general culture, the assumption that brain death constitutes actual death solves that problem. While brain function has ceased and is irreversible, the body continues to pump blood, keeping the organs alive.
For Jews, the question is whether that constitutes death according to Jewish law. If not, to remove the heart or other organs of such a person would be tantamount to murdering them, since Jewish law is clear that hastening a person's death, no matter how certain or close that death is, constitutes murder. There is, however, a debate among rabbinic decisors as to whether the secularly articulated standards of brain death match halachic parameters. Many rabbis say yes, and support organ donation (there is an Halachic Organ Donor Society that promotes this view), but others disagree, and say that death only happens with the cessation of breathing (largely ruling out organ donation, not because of a problem with the donation itself, but because by the time breathing stops, the organs are no longer fit for transplantation).
May we be protected from these kinds of issues, both the need for the organs and the ability to give organs such as those, and be blessed with health for years to come.
The vast majority of rabbis, from Reform to Orthodox, who have written about organ transplants would at least permit Jews to donate their organs upon their death. In 1989, the Chief Rabbinate of the State of Israel approved even heart transplants, which is the hardest procedure to justify because it requires recognizing whole brain death as sufficient to define a person as dead instead of the more traditional criteria of cessation of heartbeat and breath. The Conservative Movement's Committee on Jewish Law and Standards has approved a rabbinic ruling by Rabbi Joseph Prouser that claims that it is not only permissible, but mandatory to donate your organs and tissues for transplantation, for the shortage of organs has meant not only that thousands of people die each year for lack of an organ, but also that living people are called on to donate kidneys and parts of their livers. Jews may participate in such living donations, assuming that the physicians involved deem it safe for the potential donor to do so, but such operations always involve at least some risk, and so there is no requirement to do this. On the other hand, cadaveric donations, of course, involve no risk to the donor, and so Rabbi Prouser ruled that it is mandatory to make your organs available for transplant upon your death. You can see his responsum at www.rabbinicalassembly.org under the link "Contemporary Halakhah."
In asserting that it is at least permissible to donate your organs for transplant, rabbis are simply applying a central principle in Jewish law -- namely, that pikkuah nefesh, saving a person's life -- takes precedence over all but three commandments in the Torah -- in this case, over kevod ha-met, honor due to the dead body. The vast majority of rabbis would also include in this transplants that involve restoration of critical functions, such as corneas for people who would otherwise go blind or even lose sight in just one eye. After the transplant, the remaining body parts must be buried in a closed casket as usual, and because transplants must take place very soon after death, there is usually no delay in burial procedures for the donor because of the transplantation.
Most rabbis would also allow Jews to donate to an organ bank because the shortage of organs means that the organ will be used to help someone very soon even if not immediately. Whether one may donate one's body to science for purposes of medical education or research is more controversial, with only a minority of rabbis permitting it, and then only when those purposes cannot be achieved by other means. (Increasingly, first year anatomy classes in medical school are done on computer rather than on cadavers.)
Some Jews believe that you must be buried whole to be resurrected and therefore refuse to make their bodies available for transplant, but you do not want to be resurrected in the body in which you died: that did not serve you very well in the end! God is going to have to do reconstructive surgery on you in any case and can certainly replace the organs that you donate. In the meantime, it is absolutely crucial that Jews donate their organs for transplant so that we can save lives and critical bodily functions.
For more on this, see my book, Elliot N. Dorff, Matters of Life and Death: A Jewish Approach to Modern Medical Ethics (Philadelphia: Jewish Publication Society, 1998).
Judaism has no problem with receiving an organ transplant. There is some debate in Jewish law about whether one should be a donor, based on the belief that all the parts of the body must be buried in order for the dead to be physically resurrected in the time of the Messiah. As Reform Judaism does not believe, generally, in bodily resurrection, this point is moot. The Conservative movement has the most admirable position on organ donation, saying that it is a mitzvah. The Reform movement supports organ donation, but does not consider it a commandment. An interesting site for an Orthodox point of view is the Halakhic Organ Donor Society (of whom I am a card-carrying member).