Traditional candle wick materials, hemp and linen, have a good deal of loft. There are air spaces in the fibers which allow a viscous liquid like molten wax to infiltrate the material and get drawn upward ('wicking'). Cotton fibers, particularly machine spun cotton fibers, are too tight. There are not enough air spaces to draw the wax upwards. The flame runs out of uel, the wick itself burns and chars. Finally, the candle sputters out.
Cotton wicking can be made to work if it is treated first. A weak acid is used to dissolve some of the ibers so that the wax can infiltrate the material. Boric acid is normally used or this purpose. Treating cotton is not hard and can be done at home.
A boric acid solution can be made in your kitchen by combining salt water with borax (washing soda). Add one tablespoon of each to about a pint of warm water and stir to disolve. Boric acid will form when the salt and borax combines. Put a ball or skein of your wicking material (I use hand-braided perl cotton) in the solution, cap it, and leave it overnight. In the morning, take it out and let it air dry. If you soaked a skein of wicking, you can open up the skein and hang it to dry faster. When dry, drop the ball or skein in molten wax and let sit for several minutes. By pre-soaking the material in wax, your candles will light more easily and burn cleaner.
How in the world was this discovered? I am not certain and have not been able to find a lot of information about where cotton wicks were invented, but I suspect, like many advances, it was done by accident. If you wash and soak spun cotton with traditional hand-made (high borax) soap in salt water, the wicking will burn. This was probably discovered on a seacoast and then spread elsewhere.
Making candles from scratch is fun and is a bit like magic; families used to make their own lighting supplies in a big batch very year. Going through this process yourself- even once- gives you an appreciation of what life was like as well as an incredible sense of accomplishment.
