Hi Jenny, welcome to our website.
Stars have several varieties of life histories. Ordinary stars like our sun are formed by the accumulation of primordial hydrogen originating from the 'Big Bang', the beginning of our universe. There may be traces of heavier elements due to accumulation of debris of previous stellar explosions.
A young star soon gets very hot and starts to glow due to the thermonuclear reaction of conversion of hydrogen to helium. Depending on the mass of the star there are several evolutionary paths it can take. For our own sun, after an initial bright youth it will burn up all of its fuel and then slowly begin to fade out. There will still a remnant left over, you just can't see it anymore.
Other types of stars, typically much heavier than our sun, after burning all their hydrogen start to collapse due to enormous gravitational forces. A catastrophic event called a supernova can occur. The resulting leftover may be a neutron star or black hole. See:
http://en.wikipedia.org/wiki/Supernova for some gory details. In any case there is something left over but it's just not a "twinkle, twinkle, little star" anymore.
LeeH
