The Latin noun bulla, meaning “knob” or “round swelling,” is the source of a family of words starting with b followed by a vowel and the l sound (and sometimes additional letters and sounds), which are listed and defined in this post.
Ball (in senses pertaining to a round object) and related words such as ballistics are cognates of words derived from bulla; like that term, they stem from a proto-Indo-European root meaning “blow” or “swell,” though by way of a Germanic language rather than Latin. (The word for a fancy dance party, and its derivative ballet, by contrast, are from a proto-Indo-European root meaning “reach” or “throw”; though one can throw a ball that is an object as well as one that is an event, the roots are apparently unrelated.) Meanwhile, bell (and bellow) likely stem from the former root with the sense of “roar” or “sound” but are not descended from bulla.
Bill, in all the senses pertaining to a document or other piece of paper, comes ultimately from the notion of a knoblike seal used to authenticate a document. (In the sense of a bird’s beak or an ax-shaped tool or weapon, however, the word is unrelated.) Billet, referring originally to a written statement and then by extension to the housing of soldiers in private homes, authorized by such a statement, is a diminutive of bill. (Billet-doux—literally “sweet note”—is adopted from the French term meaning “love letter.”)
Bowl, and bowler (the word for a type of hat) and bowling (the term for a sport), derive from bulla, as does boll, which describes a pod of cotton produced by flowering of the cotton plant. Bollocks are testicles, and the word is British English slang for “nonsense” or a stronger retort; the spelling variant bollix is reserved for describing an act of bungling or messing up. Bolero, the word for a type of dance, comes from the extension of bulla to describe a circular motion; the name for a short jacket sometimes worn by participants in such a dance has the same origin.
Bulla itself survives in medical usage to describe a bony or blistered prominence, while bull, in the sense of a papal decree, and bulletin, denoting a notice, are cognate with bill. (Bull, when referring to male cattle and, by extension, the adult male of various species, likely comes from the proto-Indo-European root from which both “blow” and “roar” are derived, though linguists disagree about which sense inspired the word.) Bullet, meanwhile, betrays that projectiles fired by guns were originally ball shaped.