Augustus (plural Augusti) is Latin for "majestic" or "venerable". Although the use of the cognomen "Augustus" as part of one's name is generally understood to identify the Roman Emperor, this is somewhat misleading; "Augustus" was the most significant name associated with the Emperor, but it did not actually represent any sort of constitutional office. The Imperial dignity was not an ordinary office, but rather an extraordinary concentration of ordinary powers in one man's hands; "Augustus" was the name that unambiguously identified that man.
The first "Augustus" (and first man counted as a Roman Emperor) was Gaius Julius Caesar Octavianus, who was given that name by the Roman Senate on January 16, 27 BC; over the next forty years, Caesar Augustus (as he is now known) literally set the standard by which subsequent Emperors could be recognised, by accumulating various offices and powers and making his own name ("Augustus") identifiable with the consolidation of powers. Although the name signified nothing in constitutional theory, it was recognised as representing all the powers that Caesar Augustus had accumulated.
As princeps senatus (lit., "prince of the senate", "first man of the senate") he was the parliamentary leader of the house in the Senate and received diplomatic embassages on behalf of that body; as pontifex maximus (lit. "greatest bridgemaker") he was the chief priest of the Roman state religion; as bearing consular imperium he had authority equal to the official chief magistrates within Rome and as bearing imperium maius he authority greater than theirs without Rome (because of this, he outranked all provincial governors and was also supreme commander of all Roman legions); as bearing tribunicia potestas ("tribunician power") he had personal inviolability (sacrosanctitas) and the right to veto any act or proposal by any magistrate within Rome. This concentration of powers became the model by which all subsequent Emperors ruled Rome in constitutional theory (in practice this systematic and sophisticated theory gradually lost any resemblance to reality in the III and IV centuries, when the Emperors became rather more reminiscent of oriental despots than "first among equals").
Caesar Augustus also set the standard by which Roman Emperors were named. The three titles used by the majority of Roman Emperors -- "imperator", "caesar" and "augustus" -- were all used personally by Caesar Augustus (he officially styled himself "Imperator Caesar Augustus"); of these names, only "Augustus" was unique to the Emperor himself, as others could and did bear the titles "Imperator" and "Caesar" (it should be noted, however, that the Emperor's mother or wife could bear the name "Augusta"). It became customary for an Emperor-designate to adopt the name NN. Caesar (where NN. is the individual's personal name) or later NN. Nobilissimus Caesar ("NN. Most Noble Caesar"), and occasionally to be awarded the title Princeps Iuventutis ("Prince of Youth"). Upon accession to the purple, the new Emperor usually adopted the name Imperator Caesar NN. Augustus (later Emperors took to inserting Pius Felix, "Pious and Blest", and Invictus, "Unconquered", between their personal names and Augustus).
In this usage, by signifying the complete assumption of all Imperial powers, "Augustus" is roughly analogous to "Emperor", though a modern reader should be careful not to project onto the ancients a modern, monarchical understanding of what an emperor is. As noted, there was no constitutional office associated with the imperial dignity; the Emperor's personal authority (dignitas) and influence (auctoritas) derived from his position as princeps senatus, and his legal authority derived from his consulari imperium and tribunicia potestas; in Roman constitutional theory, one might consider "augustus" as being shorthand for "princeps senatus et pontifex maximus consulari imperio et tribuniciae potestate" (loosely, "Leader of the House and Chief Priest with Consular Imperium and Tribunician Power").
In many ways, "augustus" is comparable to the British dignity of prince; it is a personal title, dignity, or attribute rather than a title of nobility such as duke or king. The Emperor was most commonly referred to as princeps (basileys, "king", in Greek). Later, under the Tetrarchy, the rank of "augustus" referred to the senior Emperors, while "caesar" referred to the junior sub-Emperors. The aforementioned three principle titles of the emperors -- "imperator", "caesar", and "augustus" -- were rendered as autokratôr, kaisar, and augustos (or sebastos) in Greek. The Greek title continued to be used in the Byzantine Empire until its extinction in 1453, although "sebastos" lost its Imperial exclusivity: persons who were not the Emperor could receive titles formed from "sebastos", and "autokratôr" became the exclusive title of the Emperor.
The Latin title of the so-called "Holy Roman Emperors" was usually "Imperator Augustus", which conveys the modern understanding of "emperor" rather than the original Roman sense (i.e., the "first citizen" of the Republic). Ironically, although the German word for "emperor" is "Kaiser", a clear derivative of "caesar", that was the only one of the three principal titles of the Latin- and Greek-speaking Roman Emperors that was not regularly used in Latin by the German-speaking Holy Roman Emperors.
As a note of historical interest, the first modern use of the original sense of "emperor" was in the French Republic (République française). Napoléon Bonaparte, who was already First Consul of the French Republic (Premier Consul de la République française) for life, was crowned "Emperor of the French (Empereur des Français) in 1804; despite being ruled by an emperor, it continued to be the French Republic until 1808, when it was renamed the French Empire (Empire français).