What is behind of the attraction between a burned matchstick with a magnet?
Some matchstick heads contain iron(III) oxide as a colorant.



The yellowish color of the burning match indicates that it has low oxygen, i.e. a reducing flame. It reduces the iron oxide to iron which is attracted by the magnet.

The reduction reactions that occur are probably quite complex. Below are two simplified possible reaction equations. $\ce{(CH2)_n}$ in equation (2) stands for paraffin from the splint's impregnation. 

$$\ce{Fe2O3 + 3 CO -&gt; 2 Fe + 3 CO2} \tag{1}$$

$$\ce{Fe2O3 + \frac{1}{n} (CH2)_n -&gt; 2 Fe + CO2 + H2O} \tag{2}$$