The zipper had a long career as a gadget before it had a good name. The man who made it work called it a "separable fastener" in his patent, according to Wikipedia. Earlier versions were the "clasp locker," the "hookless fastener" and the "slide fastener." None of them sounded like something you would ask for in a shop.
The invention came in stages. Elias Howe patented a garment fastener in 1851, though it was closer to a drawstring than a true slide, and he did not market it seriously. In 1893 Whitcomb Judson showed a shoe fastener he called a "clasp locker" at the Chicago World's Fair. It jammed and drew little commercial interest. Gideon Sundback, a Swedish engineer, then produced a much sturdier design. Britannica and a 2012 Christian Science Monitor profile date his "Hookless No. 2" to 1914, while Wikipedia says the improved design first went on sale in 1913. Britannica and Wikipedia both give the patent as 1917, and Wikipedia titles it "Separable fastener." Sources differ by a year on the launch, so 1914 is the safer date.
The name came from footwear. In 1923 the B.F. Goodrich Company fitted Sundback's fastener to a line of rubber galoshes and called them "Zippers." Wikipedia says the galoshes could be closed with one quick pull. World Wide Words adds that Bertram Work, Goodrich's president, reportedly wanted "an action word" that would dramatise the way the thing zips. Britannica credits B.G. Work with giving the fastener the name.
Zip itself is imitative. World Wide Words says it was first recorded in the mid-19th century for the noise of a small object moving fast through the air, or for a tearing sound.
The name then slid from the boots to the thing on them. Wikipedia notes that Schott began using zippers on leather jackets in 1925, the first clothing use, and that makers marketed them in the 1930s for children's clothing, arguing they helped kids dress themselves. The Monitor article says that by World War II zippers were everywhere.
The word split along the Atlantic. North Americans say "zipper." In Britain the fastener is a "zip" or a "zip fastener." World Wide Words says the application of "zip" to the fastener dates only from the early 1920s.