There are two main forces at work, mechanical bonding and the van der Waals forces. Mechanical bonding refers to the liquid glue seeping into cracks and crevices in an object’s surface, then hardening to hold the objects together. The van der Waals forces refer to the attraction between molecules in the glue. the main ingredient in superglue is cyanoacrylate, an acrylic resin that can harden and cure almost instantly. All it needs to trigger the curing is water, even trace amounts from humidity in the air.
When cyanoacrylate molecules come in contact with water they form long chains, and these in turn form a strong molecular mesh as the glue hardens. This is also why the superglue sticks so well to skin, because our skin is full of tiny crevices and is also full of water, so the glue can seep in deep and still harden. This combination of a long chain molecular mesh and the speed of curing make superglue a very strong glue.
Connor Skates, Science Museum