Regarding kidney transplant
There are actually very few situations where organs can be harvested from donors. For all deceased donor transplants, the donor must be confirmed as being brain dead (both brain stem and higher cortical functions).  However, in order for the organs to remain viable they must not become ischaemic - which is obviously a huge problem when the patients heart has stopped beating.  For this reason most patients who die and go on to become organ donors were already on a ventilator in intensive care in order to keep their blood oxygenated and circulating to perfuse their organs.  

Kidneys are listed by NHSBT as a special case where a ventillator may not be required in advance to maintain viability - offhand I'm not sure why this would be - however to answer your question:

A. Neurological death - correct answer - the patient must be deceased without chance of recovery (hence brainstem and cortical death)

B. Cardiac function - were this to stop unilaterally then the patient may still be recoverable whilst the organs may be damaged due to lack of perfusion and thus not be viable for transplant

C. Respiratory function - almost identical to the cessation of cardiac function (and indeed will lead onto it very quickly)

D. Renal function - there's no point transplanting useless kidneys!

Have a look through the NHS Blood and Transplant Organ Donation site linked above, it's a great and trustworthy resource.