Concept:
Combustion is a chemical oxidation process. "Complete" combustion means all the combustible elements in the fuel (mostly Carbon and Hydrogen) are fully oxidized (turned into $CO_2$ and $H_2O$) without any unburnt fuel remaining.
Step 1: To burn completely, a fuel strictly requires oxygen (found in air) and sufficient temperature. Burning without air (A) is impossible, and low temperatures (D) lead to incomplete burning (soot and $CO$ production).
Step 2: Nitrogen makes up 79% of air, but it is inert during standard combustion. Supplying excess nitrogen (B) just absorbs heat and lowers efficiency; it does not aid burning.
Step 3: Theoretical (or stoichiometric) air is the mathematically exact, perfect amount of air calculated to provide just enough oxygen to fully oxidize every molecule of the fuel, with zero oxygen left over.
Step 4: Therefore, by definition, complete combustion is the ideal state achieved when exactly the theoretical amount of air is supplied and perfectly mixed.