The modern oil candle generally consists of a glass fuel container, with a hole at the top, with a wick holder that sits atop of the hole, through which the wick passes, into the glass container.
When the container is filled with oil, the wick and wick holder structure is replaced in the hole so the wick is in the fuel.
The fuel is then soaked up the wick, which can be made of either cotton or glass fibre. At the top of the wick, a very large surface area to volume ratio is created, wherby the fuel can begin to vapourise. When lit, it is this vapour that burns.
For this reason, if only the very top of the wick is exposed, it itself will not burn, as combustion is occurring a tiny distance above it.
If a large amount of wick is exposed, the wick itself will burn, producing a large and uncontrolled flame. This can lead to incomplete combustion and therefore some smokiness.