Instagram started life with a different name and a different job.
Kevin Systrom first built an app called Burbn. TechPP says it was named after a Kentucky bourbon, and Supply Chain Now ties it to his liking for fine whiskeys. YourStory describes it as a mash-up of Foursquare and Mafia Wars. TechPP says people used it to check in, make plans, earn points and post photos. Wikipedia says Burbn had become too similar to Foursquare, so the founders refocused it.
Systrom and Mike Krieger, who joined as co-founder, cut the extra features. What was left was photos, comments and likes. The product needed a new name, and "Instagram" was the result.
Wikipedia and other sources explain it as a portmanteau of "instant camera" and "telegram." The first half points to the instant camera, which produced a photo in seconds. The second half points to the telegram, a message sent quickly to someone else. The app combined both: take a picture now, send it now.
Another account gives the reasoning in Systrom's own terms. YourStory quotes him saying "the name sounds Camera-y," and describes the product as an "instant telegram of sorts." That is a looser rationale than the usual textbook explanation. It suggests the sound of the word mattered as much as its parts. Supply Chain Now also gives only "instant" and "telegram" and does not mention a camera.
The app launched on 6 October 2010 as a free mobile app, according to Wikipedia. The name has since outgrown both halves. Few users now think about telegrams.