The most famous of the the Punic Wars, which pitted Carthage against Rome. The First Punic War ended in a treaty, with neither side claiming victory. The second Punic War occurred because of a conflict over Spain. Carthage established a flourishing empire in Spain, and Rome, fearing this expanding empire, imposed a treaty on Carthage stating that they wouldn't expand beyond the Ebro River. Rome, however, was unable to resist accepting an alliance with a small city in the center of Carthaginian territory, called Saguntum.
In 221 BC, a young Carthaginian named Hannibal, only 25 years old, assumed control of Carthage's Iberian holdings. He initially gave a wide berth to Saguntum, not wanting to provoke conflict with Rome. Eventually, however, the Saguntines began playing politics and threatening neighboring cities with the power of their Roman allies. Hannibal, despite threats from Rome, crushed Saguntum. The Romans then demanded Hannibal be turned over to them. Carthage refused, and the Second Punic War was begun in 218 BC.
Hannibal marched the Spanish Carthaginian troops across the alps, and within months had captured all of Northern Italy. His victories brought hordes of Gauls to fight for him, and he sought to use his military strength to convince Roman cities to join Carthage. The general the Romans charged to defeat him was Quintus Fabius Maximus, who employed a strategy of avoiding open battle, shadowing Hannibal's armies and harrassing them at every turn, until they were weak enough for a full strike. The Senate grew impatient with the tactics of "The Delayer", and removed him, replacing him with two Consuls. These two promptly led the Roman army to ruin in 216 BC at The Battle of Cannae, where their force of 80,000 was crushed by the smaller army of Hannibal in a pincer movement.
After that decisive battle, Hannibal was completely unopposed in the Roman countryside. However, he lacked the men and equipment to lay seige to large cities such as Rome. The Romans then shrewdly decided to strike at his supply lines, by appointing the brilliant Publius Cornelius Scipio proconsul (unconstitutional in the Republic, as he had never served as Consul), and sending him to take the war to Carthage. By 206 he had conquered all of Spain, and in 204 he crossed into Africa and took Carthage itself (earning himself the title of Scipio Africanus). The Carthaginians sued for peace, and signed a treaty that removed Hannibal from Italy, but once he returned to Carthage, they once again took heart and Rose up against the Romans. In 202 BC, at The Battle of Zama, the armies of Carthage, led by Hannibal, met the armies of Rome, led by Scipio, and Hannibal met his first and only defeat of the war.
This was the pinnacle and defining moment of the Roman Republic. This war (and the subsequent defeat of Carthage's allies in Macedonia) left them in sole posession of the known world.