When breeding, most male birds have a swollen cloacal protuberance, the structure in the center of the photograph. A female's cloaca is, by comparison, never so swollen. Structurally, the cloacal protuberance is not a penis. The swollen cloaca is also not a cloacal phallus (an organ of ducks and ratities), which I wrote about in my 2 April 2010 post.
The cloaca is a chamber into which empty the digestive tract (the anus empties into the cloaca; the exit of the cloaca is called the vent), sexual ducts, and urinary tubes. Cloaca comes from the Latin for sewer. Male birds get the job done with a cloacal "kiss," with little or no penetration of the female. I will have more to write about cloacae (I assume cloacae is the plural of cloaca) in my next post.