A typical flame, such as that from a candle, produces light, heat, carbon dioxide and water vapor. The heat causes these combustion products to expand, which lowers their density, and they rise due to buoyancy. Fresh, oxygen-containing air can thus get into the flame, further fueling the combustion process.
Because gravity is necessary for density differences to arise, neither buoyancy nor convection occur in a zero-gravity environment such as space. Consequently, the combustion products accumulate around the flame, preventing sufficient oxygen from reaching it and sustaining the combustion reaction. Ultimately the flame goes out.
In the early years of the U.S. space program, tests were conducted on unmanned missions to ascertain what would happen to a flame in a pure oxygen environment under weightless conditions. Researchers learned that flames extinguish themselves. They ran these experiments because they hoped to have an oxygen environment for manned missions and there was concern about the possibility of a rampant fire. Unfortunately, in 1967 fire broke out in the Apollo I spacecraft while it was still on the ground and three astronauts were killed. The flames didnt self-extinguish because the launch pad was not a gravity-free environment.
Oxygen could still reach a flame in a gravity-free environment if someone blew the gas into the flame or let it "diffuse" in. It is the diffusion process that spreads the scent of a perfume in a room without air circulation: the perfume slowly mixes with the air to try to achieve a uniform distribution. This process, however, is too slow to sustain a flame.
Answer originally posted November 20, 2000.