A nova is a strong, rapid increase in the brightness of a star. The word comes from the latin for "new star," because often a star previously too dim to be seen with the naked eye can become the brightest object in the sky (besides the sun and the moon) when it becomes a nova.
Novae are now known to be caused by a star briefly re-igniting after having lain dormant for many years. Stars shine due to the nuclear fusion reactions in their cores, which process hydrogen into helium, releasing energy in the process. When the hydrogen is used up, sun-like stars slough off their outer envelopes, and become very small, very hot "white dwarfs." These white dwarfs are the inert cores of dead stars which have used up all of their available fuel. Now, stars often come in pairs, or "binaries," where two stars are in orbit around each other. If one of the stars in a binary is a white dwarf, and the other begins evolving into a red giant (a stage near the end of the life of a star, but before the white dwarf stage), the white dwarf can begin gravitationally attracting some of the gas from the atmosphere of the red giant to itself. Most of this gas will be hydrogen, and when the hydrogen reaches the surface of the incredibly hot white dwarf, it rapidly ignites, creating a large nuclear explosion on the surface of the star. This is what we see in our sky as a nova.