A round object with extra jobs
In ordinary English a ball is a sphere you can hold, kick, throw, or wind: a tennis ball, a snowball, a ball of twine. The same noun covers the object in sport and, by a short step, the sport itself when people say they are playing ball. It also names rounded parts of the body, above all the pad of the foot behind the toes, and in slang a testicle. Baseball later carved out a technical sense: a pitch that misses the strike zone. Those uses belong to one history. English also has a second noun, the formal dance, which looks identical on the page and does not belong to that history at all.
Germanic, with a missing Old English page
Middle English already had bal, later ball and balle, for a compact round body and for the thing used in games. The word is on record from about 1200. One early appearance often cited is in Layamon's Brut, around 1205, where people drive balls far across the fields.
How that form reached English is not fully settled. No Old English noun of the expected shape survives in the manuscripts. Several modern dictionaries reconstruct an unrecorded Old English *beall or *beallu, mainly because Old English did have the diminutive bealluc, meaning testicle. That diminutive is an awkward leftover unless a related word for a round mass already existed in the language. Another path is Old Norse bǫllr, also written bollr, meaning ball, which English could have taken during the Scandinavian centuries in the north and east. Merriam-Webster, the American Heritage Dictionary, and the Online Etymology Dictionary treat the lost Old English form as probable while pointing to the Norse cognate. Historical dictionaries in the Oxford tradition are more cautious: they underline that no Old English match is known and that the immediate source may be Scandinavian.
Either way the family is Germanic. Dutch bal, German Ball, Old High German ballo, and Swedish boll sit beside the English noun. The reconstructed Proto-Germanic form is *balluz, with a related weak stem *ballô behind some continental words.
French is a trap here. Later Middle English sometimes spelled the noun balle, which looks like French balle. That overlap on the page has led people to treat the English word as a borrowing from French. It is not. French balle itself is generally taken as Germanic in origin. The origin of the English word ball did not need a Romance loan to arrive; the two languages later shared a spelling that made the English noun look imported.
A root that means swelling
Behind the Germanic forms lies an Indo-European root often given as *bhel- or *bʰel-, with the sense to blow, swell, or inflate. The same root is credited with a cluster of round or swollen things in English: bowl, bole for a tree trunk, boulder, balloon, and, through the old diminutive, bollock. Latin follis, a bellows or inflated bag, is sometimes compared. If that link holds, words such as fool and follicle, which come through Latin, sit at a long distance from the same idea of something puffed out. The picture is of a shape produced by filling or swelling, not of a geometric sphere invented for mathematics.
The English noun is morphologically simple: no prefix, no native suffix on the modern form. The Old English diminutive in -uc is the only early morphological relative that actually survives in writing. Related Romance words for a ball or bale are borrowings from Germanic, not the other way around. Balloon and ballot, for example, come through Italian forms that themselves rest on that Germanic stock.
Senses that grew from the sphere
Once the round object was named, English kept adding jobs for it. By the early fourteenth century the word could mean a testicle, a sense that still rides close to the Old English diminutive. The middle of that century brings the ball of the foot and, soon after, a game played with a ball, so that the noun names both the object and the pastime. Late in the same century it is a rounded missile. Cannonballs and musket balls continue that line, and older military English could speak of powder and ball for ammunition as a pair.
Sport kept minting special uses. To have the ball, meaning to hold the advantage, is already Middle English, around 1400. Golf manuals of the late nineteenth century drilled players to keep an eye on the ball; the advice then left the tee and became a figure for attention. On the ball is early twentieth-century sporting talk for alertness. Tennis gave English the idea that the ball is in someone's court by the 1950s, a way of saying whose move it is. Baseball's ball as a called pitch is recorded from 1889, probably shortened from older compounds such as high ball and low ball.
None of these later senses needs a new etymology. They are the same round object put to work as anatomy, weapon, game, and metaphor.
The dance that is not this word
English has another noun ball meaning a gathering for dancing. That word is a seventeenth-century loan from French bal, from Old French baller, to dance, from Late Latin ballare, from Greek ballizein. The Greek verb is tied to throwing or flinging, including the athletic throwing of the body in dance. American Heritage traces it to a different Indo-European root, *gwele-, to throw, the same general line that gives ballistics. The slang sense of having a ball, a thoroughly good time, is American, from the mid twentieth century and perhaps earlier in African American vernacular, and it grows from the dance, not from the sphere. Ballet sits on that dancing line as well. Speakers collapse the two nouns because they sound the same; historically they are strangers.
What the spelling hid
English was already kicking balls across fields in the early 1200s, using a Germanic name that Dutch and German still share. Centuries later a French-looking spelling made the word seem borrowed from the Continent, and a second, unrelated French loan gave English a ball you attend rather than throw. One short noun, two genealogies, and a false French parent that never quite existed.