Praise be to Allah.
Donating organs is not all on one level. There is the donation of organs on which life depends, and there is the donation of organs on which life does not depend.
If the donation is of an organ on which life depends, such as the heart or liver, it is not permissible to donate it, according to the consensus of the scholars, because that is killing a soul.
But if it is an organ on which life does not depend, such as a kidney or blood vessels, then there is a difference of opinion among contemporary scholars, and there are two views on this matter:
1 – That it is permissible to transplant human organs
2 – That it is not permissible to transplant human organs
Fatwas stating that it is permissible have been issued by a number of conferences, seminars and committees, including: the International Islamic Conference held in Malaysia; the majority of the Islamic Fiqh Council, whose fatwa may be seen in the answer to question no. 2117; the Council of Senior Scholars in the Kingdom of Saudi Arabia; and the Fatwa Committees in Jordan, Kuwait, Egypt and Algeria.
It is also the view of a number of scholars and researchers, including Shaykh ‘Abd al-Rahmaan ibn Sa’di.
Some of the scholars favoured the view that transplants are permissible on condition that the donor is a harbi kaafir (i.e., one who is in a state of war against Islam, not one with whom the Muslims have a treaty, or who is living under Muslim protection), because the harbi kaafir has no sanctity, whereas the sanctity of the Muslim is established in life and in death.
For more information see the book Ahkaam al-Jaraahah al-Tibbiyyah by Shaykh Muhammad al-Mukhtaar al-Shanqeeti, pp. 354-391
And Allaah knows best.