The sun is too small ever to form a neutron star, which is the remnant left over after some, more massive stars explode as supernovae.
For stars like the sun, eventually, their hydrogen fuel runs out, and the stars expand into a red giant phase while helium is being fused. After several billion years, the stellar core which is left over consists mostly of carbon. Fusion stops, and the stellar core shrinks under gravity until it is about the size of earth, but contains almost all of the mass of the sun. Although this white dwarf is initially very hot and dense, over time this remnant will cool.