The term Tarquins refers to the last three kings of Rome: Lucius Tarquinius Priscus (the Elder), Servius Tullius and Lucius Tarquinius Superbus (the Haughty). These kings oversaw a period of urban expansion and development, reformed the army and strengthened Rome's military power.
The name Tarquinius was derived from the fact that Tarquinius Priscus was from Tarquinii, an Etruscan city. He was the Son of Demaratus (an aristocrat and trader from Corinth in Greece who moved to Tarquinii because he was in political trouble there) and an impoverished Etruscan aristocratic woman. He moved to Rome to pursue a public service career there as he was barred from this in Tarquinii because he was only half Etruscan and the Romans did not bar foreigners from public office.
Tarquinius Priscus defeated Latin cities which were in league against Rome and the Sabines who also attacked Rome. According to a dubious source he also defeated an Etruscan alliance. He doubled the size of the cavalry and the size of the senate. He was the first Roman king to celebrate a triumph, whose ceremonial dress and parade was identical until the time of the emperors. He also built the first wooden seating (only for the aristocracy) at the Circus Maximus. He raised the level of the forum and built the first part of the Cloaca Maxima (which at the time was a canal which drained the area) to protect it from floods.
Tarquinius Superbus was the last king. He was a tyrant and was deposed by the patricians, who then formed the republic. He put himself at the head of the Latin League (a league of Latin cities), embarked on wars, and greatly expanded the territory of Rome and the Latins. He built the temple of Jupiter Optimums Capitolinus, a massive temple on the Capitoline Hill.
In between these two kings there was Servius Tullius who is also considered a Tarquin because he was raised at the court of Tarquinius Priscus, who wanted him to succeed him. He introduced the census, reformed the army and created the assembly of the soldiers. He reformed the administrative division of the city from three to four tribes and created the assembly of the tribes. Both institutions were retained by the Republic. He also redefined the boundaries of the city, enlarging them, and completed the walls around it. Because of these radical steps, the Romans saw him as 'the second founder of Rome.'