Why does a (magic) birthday candle come on again after it is blown off?

 When a candle is blown out, normally there is a burning ember in the wick that causes a ribbon of paraffin smoke to rise from the wick. That ember is hot enough to vaporize paraffin but it is not hot enough to ignite the paraffin vapour. The key to re-lighting a candle is to add something to the wick that the ember is hot enough to ignite. That way the ember can ignite this substance and the substance can then ignite the paraffin vapour. 

The most common substance used is magnesium. Magnesium is a metal, but it happens to burn (combine with oxygen to produce light and heat) rapidly at an ignition temperature as low as 430 degrees C. Tiny bits of magnesium are mixed into the wick. While the candle is burning, liquid wax flowing up through the wick keeps the magnesium cool enough not to burn, but once the candle is blown out, the wax cools and stops rising. That lets the glowing ember heat the magnesium bits enough to set some on fire. They burn hot enough to set the paraffin vapour on fire, thus relighting the candle.