The title of Caesar, taken from the hereditary name of Gaius Julius Caesar, eventually became synonymous with the heir to the throne after the fall of the Julio-Claudians (who could trace some semblence of familial connection to Caesar the dictator).
This was actually introduced by Vespasian to tie the appointment of his sons Titus and Domitian as heirs with the original ruling dynasty.
The title of the Emperor (Imperator) was actually "Augustus" or exalted one, as well as Princeps in the early period (from Princeps inter pares Senatus for first among equals in the Senate).
Imperator was simply a military title and had nothing to do with the modern designation of the term Emperor in English. Octavian, the heir of Julius Caesar and first Roman Emperor is known by the name Augustus, as he was the first to be appointed with that title and was referred to as such by contemporaries.
This position (Augustus), essentially granting absolute power over the senate and people for life, for all practical purposes replaced the old periodic Republican title of Dictator, combining such powers as those of the Consuls and the Plebeian Tribunes. While we today know Octavian as Augustus, each Emperor was actually awarded the title.
When an Emperor or Augustus (after the Julio-Claudian period) named an heir to the throne that heir would be known thereafter as Caesar (as it evolved to become a title rather than a name like those of Tiberius, Caligula, Claudius and Nero), until his ascession, when he too would take the title Augustus.
The taking of the name Caesar originated as Augustus was named Gaius Julius Caesar like the original dictator upon his adoption. As each successive emperor was adopted (ie Tiberius by Augustus) he gained the name Caesar as an indication of this adoptive heritage.
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The most renowned member of the Julian gens (clan) was Gaius Julius Caesar. His adopted son, Gaius Octavius, assumed the name Gaius Julius Caesar Octavianus in accordance with Roman custom, later adding the title Augustus (Latin, "majestic").