Briefly, a forest in which the trees have reached their maximum natural lifespan (anywhere from 200 to 1000 years) and are in senescence, with some dead of nothing more than old age. Most of the biomass is concentrated in the ancient trees of the upper strata, leaving the soil very poor and unable to support smaller trees or undergrowth. The ecosystems of old-growth areas are in near-perfect equilibrium, remaining unchanged by natural phenomena such as forest fires and droughts.
In fact, the only thing (aside from climate change or an end of the world scenario) easily capable of disrupting the ecosystem of an old-growth forest is interference by human beings, whether by logging (the most common method), high-impact outdoor activity (which tends to concentrate the destruction in a smaller area), or introduction of an alien species, as with the old-growth chestnut forests of the East Coast, which were utterly destroyed by humans bringing the chestnut blight from Asia.
Well-known old-growth forests include the rainforests of the Amazon basin and the Tongass National Forest in Alaska.