What determines the death of a star?
The end of the nuclear fusion reactions.
The star is a big mass of gas (in general hydrogen) that is compressed on itself because of the gravitational force. When the atoms of hydrogen are close enough they start to produce fusion reactions. The reaction generate big explosions of energy that push the gas outwards. So the star is a continuos movement of gas that tend to compress because of the gravity and expands because of the nuclear reactions.
This behaviour continues for several billions of years until all the hydrogen is transformed in helium through the fusion. Then it starts to fuse the helium producing beryllium, and this process continues until the fusion produces iron. The iron does not fuse anymore because the energy required to fuse it is bigger than the energy released by the fusion process.
At this moment the only remaining force is the gravity that continue to compress the atoms, it will destroy the atomic structure, the electrons will be pushed away, then they will be captured by the nuclei that will become all made of neutrons.
If the mass is big enough, the star will curve the space-time around enough to trap even the light in what it is called a black hole. If the mass is not so big, the star will remain as a neutron star without doing anything else. In both cases the star is considered "dead".
So to answer to your question, the death of a star is due to the end of the process of nuclear fusion.