Nearly all flames are a result of a chemical reaction between oxygen and a fuel, which can be gas, wood, paper-really anything that burns. This chemical reaction releases energy in the form of heat and light. In order for this chemical reaction to happen, a critical condition is that fuel and oxygen must be in direct contact.

Once the candle flame is lit, the heat it generates melts the paraffin wax, which is the fuel in a candle. This melted wax goes up through the wick for the same reason water goes up your pants if you stand in water, a phenomenon called capillarity. The heat then causes the wax to evaporate. Nothing here involves gravity, so up until now everything should work as well in a spaceship.

The reaction uses oxygen, though, so there has to be a way by which fresh oxygen is constantly brought to the flame. On earth, this happens because hot air is lighter than colder air. The heated air around the candle is pushed upwards and replaced by colder, oxygen-rich air coming from below, resulting in air currents called convection. In outer space nothing pushes the hot, light air upwards, so there will be no convection. But convection is not the only way by which gases move about; when smells propagate, for example, they do not (usually!) create air currents. They travel by diffusion, a process much slower than convection. Diffusion tends to make the concentration of each gas uniform through space. As the candle burns oxygen, the concentration of oxygen surrounding it decreases, so the oxygen from further away diffuses towards the flame.

The question now is whether this happens fast enough to supply sufficient amounts of oxygen to the flame. If the rate at which new oxygen comes in is too slow, the flame will not be able to produce enough heat to melt and evaporate the paraffin and to activate the chemical reaction, and it will eventually die.

This is a really hard and subtle question, and I am not sure whether anyone on earth can give you a definitive argument to answer it. Apart from those scientists who decided to try it out, and went somewhere with no gravity and tried to light a candle. NASA did this both in a space shuttle and in the MIR space station. They found out that candles do burn in outer space, although more slowly than on earth. Also, since there is no convection to pull the hot air upwards, the shape of the flame is not elongated but nicely spherical. You can find pictures and details on the NASA web site.

So it turns out that the easier way to answer your question involved going in outer space and trying it out!