Not to discourage further discussion, but here is a similar question from 2014 that was answered by /u/Astrogator, whose answer I will quote here in part:

Emperor

The title Emperor is the one with the longest tradition of the three. Originally, an imperator was one who held the imperium, or the power of a magistrate. It comes from imperare, to command, rule, order. The imperator had absolute power inside his area of responsibility, borders were set by for example other holders of imperium of a higher rank, by a veto or by a majority decision of other executive bodies. As a sign of his power, an imperator was accompanied by lictores carrying fasces, a bundle of wooden rods, sometimes with an axe inside; incidentally also the origin of the symbol and word 'fascism' and still a powerful symbol of republicanism today.

Offices that brought an imperium with them were:

• Dictator, accompanied by 24 lictors outside of Rome, 12 within. His lictors kept the axe, as his powers included capital punishment even within Rome.

• Consul, 12 lictors

• Praetor, 6 lictors, inside of the ritual boundary of Rome, the pomerium, two.

• Magister equitum, the master of the horse and assistant to the dictator, 6 lictors

• Aedilis curulis, the Curule Aedils, 2 lictors, though it's a matter of debate if they always had the imperium as part of their magistrate.

From the third century BC on, it increasingly referred to a military commander, the holder of military imperium. This was a title bestowed upon a victorious commander by his soldiers, a process known as acclamation. The soldiers proclaimed him imperator, and he held that title until entering the sacred bounds of Rome, usually in triumph (if he could convince the Senate to be granted one). It was thus an incredibly prestigious title.

During Augustus' time, Imperator became part of his official name as Imperator Caesar Divi Filius, the so-called praenomen imperatoris.

Augustus

Augustus is simply a honorific bestowed upon Octavian on 17. January 27 BC, meaning 'the venerable', from augere which is related to 'augment'. His successors continued to use it as an honorific part in their official title.

Caesar

Caesar is the name of a family of the gens Iulia, which C. Iulius Caesar belonged to. Members of that family carried the cognomen Caesar to differentiate from other branches of the house. When Iulius Caesar adopted Gaius Octavius, Caesar became part of his name (he was then named C. Iulius Caesar Octavianus (Augustus)), and he continued to use it to help solidify his claim as the successor to Caesar. In turn it became part of the names of the other emperors belonging to the Iulian-Claudian line.

However, the last emperors of that line were not directly related by blood to the Iulian family, and Caesar became part of their official title and the title of all following emperors. From the time of Galba on, Caesar was also given as a title to the designated successors, by the time of Antoninianus exclusively to the eldest son/designated successor of the emperor. The reigning emperors were Imperator Caesar Augustus, while their successors were only Caesar or Imperator Caesar; thus Augustus was the title given exclusively to the reigning emperor.

Under the reign of Domitian Diocletian, this changed again as in the tetrarchy there were two superior emperors, the augusti, and two subordinate emperors, the caesares, who were also the designated successors to one of the augusti.

Vitellius broke with that tradition a bit in that he left out the title 'Caesar' for its monarchic connotations and had himself named consul perpetuus, everlasting Consul, as he wanted to associate himself more with the Republic.

So, tl;dr: Imperator was part of the official name of an emperor and some successors, Caesar was the title of emperors and their successors, and Augustus was the title of an emperor.

To quote an older answer of mine:

In the cultural orbit of the Byzantium, and especially the Slavic world, "Caesar" initially spread because Byzantine emperors conferred the title as a mark of prestige (and subordination) on various allied rulers. This practice, first attested in 695 (when Justinian II granted it to Khan Tervel of Bulgaria) became less common from the reign of Alexius Comnenus onward, when the more spectacular title "sebastokrator" came into use. By then, however, "Caesar" (Tsar) had become entrenched as a mark of prestige.

Bulgarian rulers proved especially fond of the title, partly because so many sought to dominate or conquer Byzantium.

The grand princes of Kievan Rus, by contrast, never made "Tsar" part of their official style, preferring to reserve the title for individual rulers remarkable for their strength or wisdom. After the Mongol conquests, the "Tsar" was the Khan of the Golden Horde. Ivan III's assumption of "Tsar" as an imperial title was both an appropriation of this custom and an expression of the claim that his kingdom was the natural heir of the Roman/Byzantine Empire.

In the west, "Kaiser" was the standard German title of the Holy Roman Emperors from the coronation of Otto I (962) onward. I'm not sure, however, why it was chosen over some variant of "Augustus." The prestige of the man Caesar probably had something to do with this title becoming standard (though the Commentaries were not yet widely read). It may be that no clear distinction was seen between "Caesar" and "Augustus" in terms of prestige, since both were associated with imperial rule.

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Sources

Federica Rossi, "Caesar as a political title" in A. Grafton et al. (eds.) The Classical Tradition (Harvard UP, 2010), 159-61

Oxford Dictionary of Byzantium, s.v. "Caesar"