Write the balanced chemical equations for the reaction of magnesium metal with oxygen gas, and for the reaction of magnesium metal with nitrogen gas

Magnesium element exists as single atoms. Free oxygen is always diatomic (two atoms bonded together) as O2. Nitrogen gas is also diatomic N2.

2Mg + O2 → MgO

Mg forms a +2 ion, O forms a -2 ion, so they combine and balance in a 1:1 ratio. MgO, magnesium oxide, is a solid. 
3Mg + N2→Mg3N2

Mg forms a +2 ion, but N forms a -3 ion. Hence the Mg3N2 configuration for magnesium nitride.