Is it called a cotton boll or ball?

It is known as a cotton ball with an "a" in ball.

However, while still on the plant, is called a cotton boll, as in "boll weevil," an important pest of cotton before the advent of insecticides. After the plant blooms and gets pollinated ("squaring"), the flower closes up and produces its seeds surrounded by tough fibers to protect the seeds. It is these fibers that humans harvest and transform into cotton fabric.