How does oxygen affect the burning of various materials? Would you expect the same thing to occur if no oxygen were present?

Combustion or burning of materials requires three things: combustible material (or fuel), energy (to raise the fuel to the ignition temperature), and an oxidizer (to take electrons). Oxygen is a great oxidizer and, compared to other oxidizers, helps in the release of a larger amount of energy of combustion for the same fuel.
If no oxygen is present, we will generally not observe combustion of material, unless some other good oxidizer is present. One can observe this by lighting a candle and then covering it with a glass jar. As soon as the oxygen present in the jar is consumed, the candle gets extinguished.
A similar principle is used to extinguish fires by cutting the supply of oxygen to the fuel, either through the use of water, chemicals, or sand.
Hope this helps.