There are two parts that work together in a candle: the fuel, made of some sort of wax, and the wick, made of some sort of absorbent twine. In the case of our orange, the olive oil is the wax and the wick is the central column, which supports the walls between each segment.
When you light a candle, you melt the wax in and near the wick. The wick absorbs the liquid wax and pulls it upward. The heat of the flame vaporizes the wax, and it is the wax-vapor that burns. The reason the wick does not burn is because the vaporizing wax cools the exposed wick and protects it.
wax will burn on its own, but it is like cooking oil, motor oil and coal in that you have to get it very hot for combustion to begin. An oil fire is intense and very hard to put out. Paraffin is the same way. In a candle, this works great—only the tiny amount of wax on the wick is hot enough to vaporize and burn.