Stars are huge balls of gas made primarily of Hydrogen and Helium. Nuclear fusion happens in the stars to help counter the immense gravitational field of the stars. Lighter elements fuse to form heavier elements. Hydrogen fuses to form helium, helium fuses to form lithium, lithium to carbon, carbon to neon, neon to oxygen, oxygen to silicon, and finally silicon to iron. Iron cannot fuse to give heavier elements naturally. To form other elements, the fusion reaction requires some external energy to continue. Therefore, once stars start forming iron it is inevitable that soon the star will slowly use up more and more fuel. Not long after it will lose all its fusable fuel and will no longer be able to counter the immense gravitational field on it.
Soon, the star will start collapsing in on itself. Now there are two possibilities. Either the star collapses to form a white dwarf or it collapses to form a neutron star. A white dwarf is formed when a star like our sun runs out of fuel and collapses in on itself. The gravitational field causes it to become smaller and smaller. This causes atoms to come closer and closer. Normally, these atoms are pretty comfortable around each other and don't mind the occasional bump with other atoms. However, these new conditions force the atoms to feel congested. This congestion is due to the fact that the electrons are too close to other electrons. At these large pressures, the electrons are fighting to occupy the same quantum states. This results in an outward pressure called the electron degeneracy pressure and it is enough to counteract the gravity of the star. Stars which show this continue to live on and prosper as white dwarfs.
Neutron stars on the other hand are pretty extreme. They are the remnants of the stars that at birth were about 10 - 29 solar masses (One solar mass is about the mass of the sun). These stars are so massive that on death the electron degeneracy pressure is not enough to stop the collapse of the stars. In fact in these cases, the inward gravitational force is so high that the electrons are pushed into the nuclei of the atoms, where the fuse with the protons to form neutrons. Neutrons are pretty stable and don't really want to be squished. Thus the gravitational collapsed is stopped by these neutrons thereby forming a star that many hundred of times smaller than a white dwarf but extremely dense with an immense gravitational field.
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