I hope you are asking about trick candles because you enjoy them as much as I do, not because you got annoyed when someone used them on your birthday cake.

In a normal candle the wick is a very absorbant material and transports the liquid wax to the flame, refueling it while it burns. The white smoke you see when you blow out a normal candle is vaporized paraffin (wax) from the wick. The wick still glows but isn't hot enough to ignite the paraffin vapor anymore. The reason the wick itself doesn't just burn is that the evaporating wax cools the wick. In a trick candle, or re-lighting candle, something is added to the wick. The glowing ember is hot enough to ignite the this added substance after you blow out the candle and the substance then ignites the vaporized paraffin , the candle burns again. Magnesium (a metal) is most commonly used for this purpose, because it burns rapidly at an ignition temperature as low as 950 degrees F (510 degrees C).

Now you are probably asking yourself, why the candle doesn't just explode when you add magnesium which is so reactive. Inside the burning wick, the magnesium is shielded from oxygen and cooled by liquid paraffin, but once the flame goes out magnesium dust is ignited by the ember and then ignites the paraffin smoke.

You see, there is a lot of chemistry packed into those tiny trick candles.