Glue normally fails in water because a boundary layer of water molecules sits between adhesive and surface. Mussels solve this in the byssus, a bundle of collagen-rich threads secreted by a gland in the foot. The mussel presses its foot to the rock, forms a small chamber, and extrudes protein that cures into a thread ending in a flattened adhesive plaque.
The adhesive proteins are unusually rich in 3,4-dihydroxyphenylalanine, or DOPA, a modified tyrosine. DOPA's catechol group displaces surface water and binds to mineral surfaces, and it also cross-links to give the plaque cohesive strength. Threads are made in seconds to minutes and are replaced continuously, so a mussel bed effectively re-glues itself over time.
The practical consequence is well known to anyone who has scraped a hull. The consequence often overstated is medical: DOPA-inspired adhesives have been demonstrated in laboratories and some products draw on the chemistry, but a general-purpose wet surgical glue matching mussel performance is still not routine.