Oxygen is not strictly needed, it just happens to be very, very good at what it does!
Specifically, combustion requires three things:
1) Fuel: The thing that burns. This is is often a hydrocarbon, or other organic molecule. The simplest possible fuel is pure hydrogen gas.
2) Energy: What gets the reaction started, which is true of most chemical reactions. Note that this so-called activation energy is usually much less than the energy ultimately released from combustion. This is like rolling a boulder some distance in order to roll it down a hill, which releases much potential energy.
3) Oxidizer: The molecule that accepts electrons. It turns out that combustion requires the fuel to be oxidized, that is, it donates electrons. So we need something to accept the electrons, and that's the oxidizer, which is then reduced.
Oxygen is a great oxidizer because it is so electronegative, which means it really wants to accept electrons. Only fluorine is more electronegative, and fluorine is a superb oxidizer: blow fluorine gas at nearly any substance and it bursts into flames. For example, see this video: Fluorine.
Oxygen is in many ways a perfect oxidizer for supporting life: it doesn't set things on fire like fluorine, but otherwise can oxidize very many things. When you eat, oxygen is oxidizing the food (fuel) you ingested, to generate energy you need to live. So there is "combustion" going on inside you!