Although most of what we know of the ancient Roman monarchy comes from Livy and it is likely that most (if not all) of them were fictional, studying the stories about these kings can give an insight into what the Romans themselves thought their Republic/Empire was founded upon.
These kings included:
Romulus – son of Mars, founder of Rome, and brother-killer
Numa – a peaceful king; established cults and priesthoods, reformed calendar
Tullius Hostilius – warlike, absorbed adjoining populations, expanded Rome
Ancus Marcius - improved Roman infrastructure, built first bridge, gained control of salt flats below Tiber
Tarquinius Priscus – first Etruscan king; instituted public games, drainage system
Servius Tullius – (possibly historical) created timocratic state (those with higher status have greater control in political hierarchy); built city wall; established shared cult
Tarquinius Superbus – employed many in building temples, extended Roman influence in Latium, last and most hated king of Rome
More detailed bios following:
Numa's peaceful rule was a necessary adjustment to Roman society, as Romulus' had been extremely simplistic and warlike, with a hodgepodge group of men accepted from any and all walks of life, women only acquired by force, kidnapped during a festival (the rape of the Sabine women). After Romulus' “miraculous” disappearance at death, Numa, a Sabine, was eventually chosen by the people to take over the kingship. However, he did so reluctantly. He is remembered as a wise, religious king, and one who ruled without war for the entirety of his reign.
Tullus Hostilius, on the other hand, was a warlike king, one who literally wiped Rome’s enemy, Alba Longa, from existence (and whose existence as a king was probably used as an excuse as to why there was no sign of such a group ever existing). He is remembered for expanding Rome, settling disputes among the borders, and thereafter reigning somewhat peacefully until he was struck down by the gods for not completing his religious duties well.
Ancus Martius is not as notable a king as some of the others, but is important for being the first to build a bridge across the Tiber, extending Rome’s borders to the sea, and protecting Rome from those that would attack them now that their more notably warlike king was dead. He is remembered as a good king in all respects, administratively, religiously, and in combat.
Tarquinius Priscus was the first of the last three kings of Rome, all Etruscan. He was said (by Livy) to be persuaded by his powerful wife, Tanaquil, into going to Rome and eventually getting the kingship. In his time on the throne, Tarquinius starts construction on the temple to Jupiter Optimus Maximus, drains the forum, and declares a servant boy, Servius Tullius, to be his heir (again at the prompting of his wife, because of signs from the gods). Tarquinius is then assassinated by the angry heirs of Ancus Martius, and Tanaquil rules in the shadows (pretending he is sick) until Servius can take the throne.
Servius Tullius is an interesting character for a number of reasons. First of all, he was said to have been a servant boy in the royal household, elevated to his status due to a great omen from the gods. Secondly, he is known for building the first wall around Rome, now called the Servian wall even though dating proves it to be from much later. And lastly, he started the census, an important and unique way of keeping track of the populace of Rome and their wealth. This change was carried over into the Republic, an essential tie back to earlier, better times. Servius was well-liked by both the people and the Senate, but was later assassinated in the street, his body later being run over by his own daughter’s carriage/chariot. (The story of his two daughters and the two Tarquins they would marry is a great one, but one you’ll have to look up to read for yourself, as I can’t fit it all here.)
And finally, there was Tarquinius Superbus. The last Etruscan king of Rome, whose name means proud or arrogant, was just that. Disliked by the people, Tarquinius Superbus is remembered for a number of reasons. He bought the Sibylline books, but was too ignorant to buy them all, instead allowing them to be burned until only three remained. He seized power wherever he could, making himself the sole supreme judge of Rome. Nevertheless, he did rule the military deftly and used money acquired on campaigns for public roads and works. His final downfall, however, occurred at the hands of his son, who, by raping a young and dutiful wife of one of his commanders (Lucretia), acquired all of the hate of the people of Rome. The people then overthrew the monarchy in their anger.
Thus ended the Roman monarchy.