Nomenclature
Julius and Claudius were two Roman family names; in classical Latin, they came second. Such names are inherited from father to son; but a sonless Roman aristocrat quite commonly adopted an heir, who would also take the family name - this could be done in his will. Thus (Gaius) Julius Caesar adopted his sister's grandson, Gaius Octavius, who became a Julius, eventually named Imperator Caesar Augustus, normally called in English Augustus, the founder of the Empire. The next four emperors were closely related through a combination of blood relation, marriage and adoption.
Tiberius, the eldest son of Augustus' wife Livia by her first husband, was a Claudian by birth. He later became Augustus' son-in-law after marrying the emperor's only daughter, Julia the Elder. Tiberius, like Augustus before him, was adopted into the Julii upon becoming the emperor's heir apparent.
Caligula had both Julian and Claudian ancestry, making him the first actual "Julio-Claudian" emperor. Through his mother, Agrippina the Elder, Caligula was a great-grandson of Augustus. His father, Germanicus, was a scion of both Julii and Claudii, descending from the respective bloodlines of Octavia Minor and Livia Drusilla. After Tiberius' adoption by Augustus, Tiberius was required to adopt Germanicus, thus giving the Julius nomen to the members of Caligula's family.
Claudius was a Claudian through his father, Nero Claudius Drusus, and also possessed a blood connection to the Julian branch of the Imperial Family through his mother, Antonia Minor. After becoming emperor, Claudius added the cognomen Caesar to his full name.
Nero was a direct descendant of Augustus and Livia through his mother, Agrippina the Younger. The Julio-Claudian Dynasty collapsed after his reign ended through suicide.
Read more about this topic: Julio-Claudian Dynasty