How does a candle burn?

A candle is a neat little self-feeding system. When you light the wick, the heat melts the solid wax near the flame into a small pool of liquid. That liquid wax is pulled up the wick by capillary action, the same effect that draws water up a paper towel. Near the hot flame the wax turns from liquid into vapour, and it is this wax vapour, mixed with oxygen from the air, that actually burns. The burning releases heat, which melts more wax and keeps the whole cycle going until the wax runs out. The wick itself mostly just delivers fuel; because it stays coated in wax, it burns away only slowly. The flame glows yellow because it is full of tiny soot particles heated until they shine.