The first lollipops had no sticks. When the word turns up in 1784, as "lolly-pops," it names a soft, coarse sweet made of treacle and sugar, usually with butter and flour added.
Where the word came from is not known. Etymonline, following the Oxford English Dictionary, calls it "of obscure formation" and offers two guesses. The first half may be loll, "to dangle," a verb used of a hanging tongue since the 1610s. Or it may be the northern English dialect word lolly, meaning "the tongue." The second half may be pop in the sense of a strike or slap.
If either guess is right, the name describes eating the sweet rather than the sweet itself. It is something you work at with your tongue.
The meaning moved slowly toward the modern candy. By 1849 "lollipop" could mean anything sweet but insubstantial. The short form "lolly" appears in 1854. Hard candy on a stick is the 1920s sense in Etymonline's dating.
That last step is where an American company comes in, along with a popular myth. George Smith was a co-owner of the Bradley Smith Company of New Haven, Connecticut. Connecticut History says the company began producing its "Lolly Pops" in 1908. The name "Lolly Pop" was officially registered to the firm on October 13, 1931. Wikipedia says the Patent Office first turned the application down after finding "lollipop" in an early 19th-century English dictionary.
The same site says the name "was reportedly inspired by that of a racehorse Smith had seen at a local fair." That may be true of his brand. It cannot be where the word comes from. "Lolly-pops" are on record more than a century before Smith, and Wikipedia notes that sweets on sticks and the word itself both existed earlier.
What Smith's company may deserve credit for is the modern sense. Connecticut History calls him the first to apply the term to hard candy on a stick. The word was too common to stay private, though. Wikipedia says the company went bankrupt in the Great Depression and lost the trademark. Connecticut History puts it more simply: the term had become so widely used that the trademark could not be maintained.
The tongue theory has relatives. Loll also produced lollop, an extended form from 1745, and the 1660s insult lollpoop, a lazy, idle drone. In mid-20th-century British slang, lolly went on to mean money.