The word ampersand was made by children mumbling the alphabet.
For a long time, & was taught as part of the alphabet, recited after Z. Wikipedia says it was regarded as the 27th letter of English as taught to children in the US and elsewhere. That created a small problem at recitation time.
Letters that were also words on their own got special treatment. Pupils marked them with the Latin per se, "by itself." The & got the same handling, so the alphabet ended "X, Y, Z, and per se and." The phrase meant that the sign, standing alone, reads "and." Etymonline adds that the phrase set the lone symbol apart from its use in abbreviations such as &c. for et cetera.
Said fast, over and over, the phrase wore down into one word. Grammarphobia, citing the Oxford English Dictionary, gives the earliest spellings as "ampuse and" in 1777 and "appersiand" in 1785. Later came "ampus-and" in 1859 and "Amperzand" in 1869. Etymonline dates ampersand itself to 1797 and records an earlier colloquial form, ampassy, from 1706. Wikipedia says the word was in common use by 1837.
The symbol is far older than its name. It began as a ligature of the letters e and t, spelling the Latin et, "and." Wikipedia traces the joined form to Roman cursive in the 1st century CE. Grammarphobia points to the earliest known example: graffiti at Pompeii, preserved by the eruption of Vesuvius in AD 79.
The & did not come from the Tironian notes, the Roman shorthand system. Etymonline says those used a different sign for "and," shaped like a 7. That sign survives today in Irish and Scottish Gaelic, where it does the job English gives to &.
One popular story is false. It holds that the French physicist André-Marie Ampère used the symbol so often in his writing that people called it "Ampère's and." Wikipedia labels this claim false. The name came from a nursery drill, not a scientist.
The sign's spot at the end of the alphabet gave it one more meaning. Etymonline notes that by the 1880s ampersand was slang for the backside, because it came last.