Yes stars die, but it usually takes billion years for that to happen. It occurs when they lose all their hydrogen and helium, hence left with no fuel to burn.
Two possible reactions can happen depending upon the size of the star.
1) The red giant
When stars like the sun which are comparatively smalle, burn out, their core begins to contract. The few helium that is left starts heating up the core, this further causes the outer layer to heat up, this in turn causes the outer layers to expand. The sun might even reach up to the level of the earth's orbit if sun turns into a red giant. Finally when all the helium is used up, their atoms get fused to carbon, leaving a cloud of waste around it (nebula). Soon the red giant shrinks to a white dwarf, later a black dwarf.
2) Supernova
The bigger stars die sooner than the smaller and medium sized stars because they have more fuel. After the helium is gone, the mass is enough to fuse carbon into heavier elements such as oxygen, silicon, sulphur and iron. The inner iron cores shrinks, as the core shrinks all the electrons and protons fuse into neutrons. This causes the core to further shrink, if the iron core is in the size of the earth, it is become a sphere of 6 miles within a few seconds. The entire mass of the star collapses internally causing a big explosion called a supernova. This might cause a black hole or a neutron star depending on the size of the star.
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