Nova is a Latin word meaning new, and it describes the appearance of a seemingly "new star" in the sky, a brilliant object in a place where there was previously only a very faint star, or perhaps nothing at all.
A nova is a phenomenon that happens in a binary star system containing a white dwarf and a stable companion star. A white dwarf is the dead, collapsed core of a star that formerly was about the size of the Sun. When the Sun dies, it will become a white dwarf. Unlike the Sun, however, many stars exist in binary systems, where two stars orbit one another. In many binaries, these stars may be separated by a distance much less than the distance from the Sun to Earth.
Suppose we have a binary system with a white dwarf and a companion star that is expanding to become a red giant star. As the surface of the red giant star expands, it gets progressively closer to the white dwarf. Eventually, the surface of the giant may reach a critical point between the two stars where the gravity of the white dwarf is actually stronger than the giant's own gravity. If this happens, matter will begin streaming off the giant's surface and onto the white dwarf. This is like overfilling a bucket with water—eventually the water will overflow, and if there is an adjacent bucket, begin pouring into it. Likewise, the large star will begin to lose matter once its surface expands past its Roche lobe, the imaginary surface beyond which the giant star's gravity is no longer sufficient for it to retain its matter.
The white dwarf has tremendously strong gravity, because it is very massive and very small. Therefore, the companion star's matter, which is mostly hydrogen, is squashed into a dense, thin, hot layer on the white dwarf's surface. The more matter that streams onto the white dwarf, the hotter it gets, and eventually, thermonuclear fusion reactions begin. These reactions are just like those that occur in the center of a stable star like the Sun, converting the hydrogen to helium with an accompanying enormous release of energy. In a brief but violent cataclysm, the hydrogen on the white dwarf's surface burns away, and while it does, the white dwarf brightens by as much as a factor of a million (15 magnitudes). This is a nova, and after reaching its peak brightness, it slowly fades over a period of weeks to months.
Because mass transfer in a binary system does not stop after a nova explosion, the white dwarf will start to reaccumulate matter. Novae therefore are recurrent, with the length of time between nova outbursts in a system depending on how fast the companion star is losing matter to the white dwarf. If the stream is just a trickle, it might be thousands of years until the next outburst. Other novae recur much more frequently. As a single star, the Sun is unlikely ever to become a nova after it dies. It may accrete enough matter just from the interstellar medium to become a nova, but such novae are extremely rare events due to the low rate of accretion of matter.
Novae should never be confused with supernovae, which are not just "big novae." Supernovae involve the explosion and destruction of a star or a white dwarf, while a nova is merely the conflagration of a surface layer of hydrogen on a white dwarf. Novae are much more common than supernovae, and they do not release nearly as much energy. Nevertheless, they are a good reason to be familiar with the sky: things do change up there, and you never know when you might look up and see a familiar constellation looking a bit different. Just this kind of thing happened as recently as December, 1999, when a bright, naked-eye nova appeared in the constellation Aquila, the Eagle. At its maximum, it was as bright as many of the stars in the constellation, and for a few days at least, viewers were treated to the spectacle of a truly "new star" in an otherwise familiar constellation.