Cyanoacrylates include methyl 2-cyanoacrylatecommonly sold under the trade names "Super Glue".In general, cyanoacrylate (consists of monomers of cyanoacrylate molecules) is an acrylic resin that rapidly polymerises in the presence of water (specifically hydroxide ions), forming long, strong chains, joining the bonded surfaces together. Because the presence of moisture causes the glue to set, exposure to normal levels of humidity in the air causes a thin skin to start to form within seconds, which very greatly slows the reaction. Because of this cyanoacrylate is applied thinly, to ensure that the reaction proceeds rapidly and a strong bond is formed within a reasonable time.
So to sum up, in order to start the reaction off some water is normally needed.. damp things stick better/quicker than dry ones and the glue goes hard faster on a humid day.
In cotton wool, which is made of cellulose, a polymer of sugar molecules, there are lots and lots of hydroxy (-OH or alcohol groups), which can start the reaction in the same way as the water does, only because there are lots of them they can start many more reactions at once.
Since the reaction gives out heat, the cotton bud therefore gets hot (and as it becomes hotter so the reaction goes faster etc), and it may get hot enough to catch fire.