Julius Caesar's praenomen, the first part of his Roman name, was Gaius. His full traditional name was Gaius Julius Caesar: Gaius was the personal name, Julius identified his gens or clan, and Caesar was his cognomen, marking a family branch.
Roman names did not work exactly like modern first names and surnames. The praenomen was usually given early in life and came from a limited set of traditional names. Gaius was one of the most common Roman praenomina and was often abbreviated as C, a survival from the older spelling Caius.
The name is sometimes written Caius in English-language works, but Gaius is the standard scholarly form and the stored answer is correct. The letter G was introduced into the Latin alphabet in the third century BC, yet the abbreviation C continued to be used for Gaius. That is why Roman inscriptions can produce apparently different spellings.
The middle element, Julius, referred to the patrician Julian clan. Caesar later became an imperial title because his adopted heir, Gaius Octavius, took the name Gaius Julius Caesar and became Augustus.