A nova is an eruption that occurs on the surface of a white dwarf in a double (or binary) star system. A white dwarf is essentially the ember of a dead star — it's the hot, dense, leftover remnant of a low-mass star that has run out of fuel to sustain nuclear fusion. With no available fuel left, fusion ceases and the star is no longer continuously producing energy.
If the white dwarf is in a binary star system, and if the other star in the binary is close enough, the white dwarf can gravitationally pull hydrogen gas from the outer layers of the companion star onto its surface. The gas builds up in a layer on the surface of the white dwarf until it gets hot and dense enough at the base of that layer for runaway hydrogen fusion to ignite.
The fusion of all of that hydrogen releases a massive amount of energy, causing the system to quickly get much brighter, which is what we call a nova! In most stars, this only occurs once every 10,000 or so years, but in some special systems, the nova eruptions happen much more frequently, recurring on a timescale of less than about 100 years, hence the name recurrent nova.