1) Simple diffusion. For smaller animals or animals with very thin body walls (such as sponges, cnidarians, etc.), oxygen in the water simply enters the cells. No extra organs needed. This doesn't work for larger organisms.
2) Organ-based diffusion. This is what gills do. Gills are basically complexly branched organs that provide a huge surface area for oxygen to diffuse into the bloodstream. Because the dissolved oxygen in the water is far less than that in air*, water-breathing organisms generally cannot power large brains as can land-based organisms.
3) Air breathing. For cetaceans, pinnipeds, and other sea-dwelling non-fish, breathing water is simply not an option. The mechanisms for breathing are far too deeply buried for them to re-evolve to breath water. So these animals have various methods for being really good at holding their breath.
*'dissolved oxygen' means FREE oxygen. You might remember from chem class that water is made of two hydrogens and one oxygen, so you'd thing the animals in the water would be literally surrounded by oxygen. But this water-bound oxygen is NOT free, and it actually would require a substantial energy INPUT to free it.
Gills
They have gills which take in water. Then they dissolve the oxygen in water and use that oxygen.
