Velcro is a brand name built from two French words, and it began with a weed stuck in a dog's fur. Its owners now spend real effort asking people not to use the word.
The weed was burdock. In 1941 the Swiss engineer George de Mestral came back from a hunting trip covered in its burrs, and so was his dog. Instead of just picking them off, he looked at one under a microscope. He saw hundreds of tiny hooks that caught on anything with a loop, such as fabric or fur.
Turning that into a product took years. De Mestral filed for a Swiss patent in 1951, and it was granted in 1955. The fastener had two sides: one covered in small hooks, the other in soft loops.
The name describes those two sides. De Mestral blended velours, French for "velvet," with crochet, French for "hook." The velvet presumably stands for the soft loop side, and the hook for the other. Etymonline reads it as velours croché, "hooked velvet." Either way, it is the first syllable of each word joined together, a trademark rather than a word that grew naturally.
According to the company, the first trademark registration came in 1956. Etymonline records Velcro in English by 1958, when Popular Science described "a nonmetallic fastener with no mechanical parts." The American business was based in Manchester, New Hampshire.
A popular myth says NASA invented it. It did not. Wikipedia notes that the fastener spread widely after NASA adopted it, which is probably where the idea came from.
The name's success became a problem. People used "velcro" for any hook-and-loop fastener, and even as a verb. A trademark used that way can become generic, which weakens the owner's rights. In 2017 Velcro Companies released a music video, "Don't Say Velcro," performed as if by its in-house legal team, singing the request. The company asks people to say "hook-and-loop" for the generic thing. Its guidelines state that the mark is "not a verb, a noun, or a stand-alone word."
So the word exists because of a burr, and it survives partly because its owners keep asking the rest of us to stop using it.