Fires on earth are anchored by gravity, but the combustion gases are hot and light, so they rise. As the flame goes up, more air is sucked into the base of the fire, feeding more oxygen to the fire and making it burn more strongly.
In space, where we have little or no gravity, there is nothing to make fires go up, and the fire has a harder time obtaining a supply of oxygen. In the microgravity of the orbiting space shuttle, oxygen molecules can only get to a fire by either being pushed into it by something like a fan--which would take the place of the suction of air into the fire on the earth--or by diffusing through the fire gases, much like ink or oil spreads out on the surface of water.
The diffusion process is slower than the suction created by flames on earth. The result is that the combusting gases also have to diffuse outward to obtain new oxygen, so the fire becomes bigger. But as its area grows in size, more heat is lost through radiation, just as heat radiates from a fireplace or campfire. If enough heat is lost the burning material will be cooled below its ignition temperature, and the fire will go out. This usually doesn't happen on the earth because air is drawn in fast enough to supply the fuel.
Fires on a spacecraft would most likely be the result of an accident, such as an overheated electrical wire. Lighting a fire where there is little or no gravity would be easier than on the earth, but once it gets going it would be less of a threat than on earth.
"By studying the way things burn in a new combustion laboratory carried on the space shuttle we can learn more about how to prevent and control fires on earth. We are also learning how to utilize heat and combustion in the difficult conditions of space where they will be essential for manufacturing materials, such as ultrapure semiconductors, in orbit."