A - Thanks for the question. First of all, it is a heroic thing to do for a Catholic, or anyone else, to volunteer to donate organs after their death. Here is what John Paul II wrote about it in the context of talking about gestures that build up a culture of life:
A particularly praiseworthy example of such gestures is the donation of organs, performed in an ethically acceptable manner, with a view to offering a chance of health and even of life itself to the sick who sometimes have no other hope (Evangelium Vitae, 86).Some of the other considerations we must think about when this issue arises is the freedom to give the organs to another. Therefore, the dead person must have given consent or had another with authority to do so give consent for them. It cannot be a forced harvesting of their body's organs.
Also, if the person is alive and giving an organ willingly (such as one of their kidneys) then they must not do so if it causes damage that might lead to their death (e.g., giving your only kidney).
The Catechism says this about organ donation:
2296 Organ transplants are in conformity with the moral law if the physical and psychological dangers and risks to the donor are proportionate to the good sought for the recipient. Organ donation after death is a noble and meritorious act and is to be encouraged as a expression of generous solidarity. It is not morally acceptable if the donor or his proxy has not given explicit consent. Moreover, it is not morally admissible to bring about the disabling mutilation or death of a human being, even in order to delay the death of other persons.Lastly, we must make sure that if an organ is being harvested from a body, that the person is dead. This leads to the question of death. At what point do we define death? This is both a philosophical and a medical question. The Church is only competent in answering the philosophical part - when the soul and body are separated.
The medical question is out of the realm of the Church's expertise. Therefore, the Pope left this question up to doctors and scientists to answer. But, within certain guidlines and limits of science and medicine. So, JPII said the following to a group of transplant doctors in 2000:
“Vital organs which occur singly in the body can be removed only after death; that is, from the body of someone who is certainly dead ... the death of a person is a single event consisting in the total disintegration of that unity and integrated whole that is the personal self ... The death of a person is an event which no scientific technique or empirical method can identify directly ... the ‘criteria’ for ascertaining death used by medicine today should not be understood as the technical scientific determination of that exact moment of a person’s death, but as a scientifically secure means of identifying the biological signs that a person has died.”
Therefore - CERTAIN death is necessary. How do we define this medically?
Total and complete death of the brain is the answer. Why? Because without it we cannot be certain. The National Catholic Bioethics Center is a great resource for all bioethics issues. I highly recommend them. For this issue, you can read their FAQs about brain death here.
I hope this helps.