Augustus' great achievement was the restoration of law and order after years of late Republic and early Empire civil war. It was he, rather than Julius Caesar, who really established the Empire. But although he had imperial power (he was legally granted "imperium"), he made a point of always acting only as "Princeps", the "first among equals". By doing so he publicly acknowledged the role of the Senate and great families and thereby avoided the mistake that had cost his arrogant uncle, Julius Caesar, his life. His success resided most clearly in his longevity: by the time he died in 14 AD, he was 77 years old, and most living Romans had no experience of any form of government other than his Principate. It was this that allowed the Principate system that he had set in place to survive the unruly brood of "Julio-Claudian" emperors that succeeded him.
Augustus held personal power granted freely by the senate and people of Rome. Because he did not want to look like a king or like that arrogant uncle, he had not really set up a succession. He did groom -- but outlived -- several successors before settling on Tiberius. After Tiberius came Gaius (Caligula), Claudius, and Nero in a quick slide through quirky to insane. In 68 AD Nero committed suicide (thus barely avoiding being torn to bits by the mob) and was followed by four military emperors in quick succession and finally by Vespasian, the first of the Flavian dynasty of Emperors. (Sixty-nine AD was known as the year of the four Emperors: Galba, Sabinus, Vitellius, Otho, before reaching Vespasian who had a real multi-year run.)
The Internet has the whole story, of course in much greater detail. A very accessible short modern biography of Augustus is at http://www.roman-emperors.org/auggie.htm. If you must have the original ancient Roman sources, they are available in English and Latin at http://www.fordham.edu/halsall/ancient/asbook3.html (scroll about one fourth of the way down the page), but the ancient historians, frankly, were not as good as modern ones.
PS: Many biographies of other important Roman imperial persons are available at http://www.roman-emperors.org/impindex.htm (modern historians) and at http://www.fordham.edu/halsall/ancient/asbook3.html (mostly ancient historians).