In microgravity, flames behave very differently than on earth due to a lack of buoyant forces. On earth, a flame can continue burning because, as the warm air around it rises, cooler air gets entrained, drawing fresh oxygen to the flame. In microgravity, both the heat from the flame and the oxygen it needs to burn move only by molecular diffusion, the random motion of molecules, or the background environmental flow (air circulation on the ISS, for example). This video shows a test of the Flame Extinguishment Experiment (FLEX) currently flying onboard the ISS. A fuel droplet is ignited, burns in a symmetric sphere and then eventually extinguishes either due to a lack of fuel or a lack of oxygen. Check out this NASA press release for more, including great quotes like this:
“As a Princeton undergrad, I saw in a graduate course the conservation equations of combustion and realized that those equations were complex enough to occupy me for the rest of my life; they contained so much interesting physics.” – Forman Williams