Marcus Aurelius, unlike the other Good Emperors, had a son. His son's name was Commodus, and when Marcus Aurelius died in 180 AD Commodus took over the Roman Empire. But Commodus, like the later Julio-Claudians or like Domitian, had grown up at court, and liked partying more than he liked fighting or working at running the Empire. Still he did well enough at first, and made peace with the Germans. But when he came back to Rome, his sister Lucilla tried to kill him, with the help of some Senators. Even though the plot failed, Commodus became very suspicious (like Domitian again!) and had a lot of Senators and other people killed. One plan he had to kill some of his closest friends backfired when they killed him instead in 192 AD.
But these friends didn't have anybody in mind to be the next emperor. A few of the more powerful men in Rome called themselves emperor, but all of them were quickly killed in their turn. Civil war seemed unavoidable.
He beat the other candidates, though it took him several years to finally defeat the most serious threat, Clodius Albinus, also an African and the governor of Britain.
Then in 197 AD the Parthians, seeing civil war in the Roman Empire, attacked again. Septimius Severus went there and pushed the Parthians back again.
After travelling around the Empire, he spent four years in Rome before going to England to fight an invasion there. Septimius Severus died in England, at York, in February 211 AD. He left the Empire jointly to his two sons, Caracalla and Geta. He is said to have told them to take care of each other and the army, and never mind anything else.
The Ancient Roman World, by Ronald Mellor (2004). Straight political history, For teens.
Classical Rome, by John Clare (1993). For kids, the whole political history from beginning to end.
The Romans: From Village to Empire, by Mary Boatwright, Daniel Gargola, and Richard Talbert (2004). Okay, it's a little dry, but it is up to date and has all the facts you could want.
The Roman Empire, by Colin Wells (1984). More readable. Alternates chapters on political and social history. Unfortunately, he stops at the third century crisis.
The Severans: The Changed Roman Empire, by Michael Grant (1996).
Septimius Severus, by Anthony Birley (1971). Emphasizes the emperor's African origins.