What is the Catholic Church teaching on organ donation?
This is really a great question. For simplicity, let's split the answer into 2 categories - organ donations from a living person and those from a dead body.
Organ Donations From A Living Person
The transplantation of organs can be seen as an act of extreme love. Transplanting from living donors is morally permissible when the donation will not sacrifice or seriously impair any essential bodily function and the anticipated benefit to the recipient is proportionate to the harm done to the donor. So, for example, if Bob has two perfectly good kidneys, it would be morally acceptable to donate one of them so that someone with no functioning kidneys could live a normal life. It would not be acceptable for Bob to donate both of his kidneys.
Also, the freedom of the will of the donor must be respected. For example, back to that kidney. If Bob is giving the kidney under duress, this would not be considered morally okay. Finally, it is not morally acceptable for the donor to accure an economic advantage by selling his or her organs. A donor may be reimbursed medical expenses and lost wages, for example, but the Church deems the commercializatoin of organ donation as immoral.
Organ Donations From A Deceased Person
Generally speaking, organ donations from a deceased person are moral. However, most organs deteriorate very quickly after death, so there is often pressure from the medical community to remove them as soon as possible. If vital organs are removed before a person dies, thus contributing to their death, this is not acceptable from the position of the Church that defends a person's human dignity and right to life.