His contemporaries called him Gaius -- though not normally to his face. It seems that his own preference for a short-form name was his father's cognomen, 'Germanicus', but it never got traction because his father was too famous and Gaius' reign didn't last long enough.

The nickname Caligula was given him when he was a toddler because his parents dressed him up in a miniature uniform and the soldiers found it adorable.

Caligula was only normally used of him as a child, and only by soldiers from his father's army. Historians didn't start using the name until 300 years after his death.

Here's a more detailed write-up from a few weeks ago, and a still more detailed piece that I wrote on another site a couple of days later.