It depends on the species. A few birds are truly monogamous, including some sea eagles, macaws, crows, and ravens - they generally mate only with one individual, keep the same mate from year to year, and look for a new mate only if their previous mate is killed or incapacitated. Many species, including most passerines, hawks, and woodpeckers, choose a mate for a season, but change mates from year to year. Some small birds (e.g., English sparrows, other small passerines) choose a mate, but will mate with another bird as well if the opportunity presents itself.
At the other extreme are species where one male mates with many females (phesants, grouse, bowerbirds, ducks, hummingbirds, birds of paradise, etc.) or where females may mate with several males (eclectus parrots).