Besides a kidney, are there any other organs that you are allowed to donate?
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Well according to Wikipedia, here's the list of organs that can currently be transplanted from a living donor. For something like a kidney donation, the donor has two and gives one. For something like a liver donation, they take a piece from the donor, which he can live without (and will be enough to help the recipient):
As far as I know, the halachic logic is the same for all of these: you're putting yourself at "some risk" (safek sakana) to save the life of someone in "serious mortal danger" (vadai sakana). (Okay with an ordinary blood transfusion your risk is quite minimal. With something like bone-marrow, from what I hear there's also a question not so much of risk but of pain.) The common psak l'halacha is that the Yerushalmi requires one to put himself in some risk to save a life, but we follow the Babylonian Talmud, which allows it and considers it a great mitzva. One exception would be if the donor risk exceeds the recipient's risk. Another exception would be certain cases, if the procedure has a <50% success rate: as long as the recipient could get off the operating table without this transplant, he's considered "presumed alive" and we try to save his life. But if, for instance, they've already taken out his heart and are about to put in a new one, he's no longer "presumed living", and thus we need 50+% chances. (This was R' Untermann's ruling regarding heart transplants from dead donors; the success rate has since exceeded 50%.) A last exception would be that as it's permitted but not required, if the donor's family objects or there's similar reason weighing against doing so. But basically if I understand the current numbers, all of these should be allowed.
As for donations from a dead body, we allow anything to be done to a dead body to save a life (see above caveats); the bigger concern is which organs get harvested before halachic death.
If I recall, Newsweek had a detailed description of a living-donor liver transplant, circa 2002. The recipient had damaged his liver from a youth full of drugs and alcohol, but had been clean for many years. I asked Rabbi Welcher if we factor into our decision whether the person in danger brought it upon himself like this case, he said probably not.
One issue dealt with by pos'kim is the prohibition against wounding oneself. Rabbi Moshe Feinstein in a letter reprinted in Igros Moshe (Choshen Mishpat Ⅰ 103) says that it is permissible to donate blood even though there is no known recipient now, in a case where the donor gets paid for his donation; part of his reasoning is that bloodletting, long thought to benefit the bloodlettee (or whatever the word is), probably does in fact benefit him, albeit slightly. Note that Rabbi Feinstein was dealing with the case he was presented (viz, where the donor is paid, etc.), and should not necessarily be construed as forbidding other cases. As always, CYLOR.
It is the opinion of a growing number that it is such a mitzvah to save a life that a person who is dead can donate any organs. Please see the Halakhic Organ Donation Society for more information.
The definition of halakhically "dead" is disputed - some see it as the complete cessation of the heart-beat and others the complete cessation of independent breath. The latter definition (aka, "brain stem death") is held by US and Israeli law and allows for more organs to save others' lives.
From that site: