When and what organ donation is permitted?
The question is a little too general to be dealt with in this forum. Very generally, the main problem with organ donation is that for major organs (heart, lungs, liver), there is a chance that the organ will be taken while the donor is still halachically alive, implying his halachic murder. This is the reason for which many object to organ donations on religious grounds. However, in the State of Israel the Rabbinate has become involved in the matter, and arranged for donations to be performed only after the halachic death of the donor has been verified. Not all rely on this arrangement, but many do. Another issue is the question of being buried with all one’s organs intact, which is important in the kabbalistic tradition, though this issue would not be weighty enough by comparison with the potential saving of a person’s live by means of the donation.
With regard to more minor organs, it is certainly permitted to donate a kidney, if the donation might be able to save somebody’s life, and this operation is often performed, as are other donations of lesser importance. With regard to ovarian and testicular transplantations, the question of sirus (castration) must be raised. In addition, the question of motherhood and fatherhood arises, in particular in ovarian transplantation, and this question remains generally unresolved.
Sources: For a good overview of the issues involved in major organ donations, see http://www.agudatadi.org.il/ShowPage.asp?pid=91, which lists most of the sources that have discussed the questions of organ donations in the recent era. For an overview of ovarian transplantation, see http://olamot.net/ratz-katzvi/%D7%94%D7%A9%D7%AA%D7%9C%D7%AA-%D7%A9%D7%97%D7%9C%D7%95%D7%AA-%D7%91%D7%99%D7%A8%D7%95%D7%A8-%D7%94%D7%9C%D7%9B%D7%AA%D7%99.