The founder of the app, Kevin Systrom, was interested in photography before he created Instagram. He really loved the instant photos made by Polaroid, for example. This is where the word “instant” came from. Photos can also be sent as messages or telegrams, hence the combination of instant and telegram to make Instagram.
The logo of the company, in turn, was inspired by the name: a slightly altered picture of the retro camera Polaroid OneStep. Later, it was changed to be more minimalistic but you can still see the camera in the logo.