The asymptotic giant branch (AGB) is a portion of the H-R diagram (plotting stellar luminosity against temperature) forming a branch line stemming from a line designating stars in their main sequence phase. As such, the AGB is a phase in the life of some main sequence stars, which includes mid-sized stars such as the Sun.
Stars completing their normal long-term hydrogen burning (the main sequence) go into a red giant phase, burning hydrogen in a shell around the helium core formed through its earlier life. Following this is the horizontal branch phase, with the a helium burning core and hydrogen burning around it.
After this is the asymptotic giant branch: helium fusion has produced a core of carbon and oxygen and the helium only burns in a shell around this core, with continued hydrogen burning outside the helium shell.
In some cases, when the AGB fusion has slowed due to running low on helium, there are relatively brief flashes of helium fusion, the star being known as a born-again star. This lasts only a short while, on the order of two hundred years.
The term is also used for other situations where a dying star brightens for more than a "flash".