The year was 218 B.C. The Roman Republic at this point ecompassed all of Italy, including Corsica, Sicily, and Sardinia. Those three islands were won in the First Punic War, a massive war raged from 264 - 241 B.C. against the Mediterranean's greatest power, the Carthaginian Empire. After an exhuastive war in which Rome was forced to learn the arts of naval warfare (and mastering it), the stubborn Romans were able to beat all odds and defeat Carthage.

After the war, while Rome was rolling around in its new position as the strongest city in the Mediterranean, Carthage was gearing up for the next battle. Hamilcar Barca, a veteran general of the First Punic War, was annexing territory in Spain to use for extraction of wealth (Spain had lots and lots of silver) and as a base of manpower (Carthage used mostly mercenaries in its wars, not Carthaginian citizens). Hamilcar had brought his son, Hannibal, who took over the campaign after Hamilcar's death.

Hannibal displayed his brilliance in his wars against the Spanish tribes, eventually encroaching upon a Roman ally. Rome asked Hannibal to stop, and when he did not, Rome went straight to Carthage. The Carthaginians opted for war.

Hannibal with a massive mercenary army (including his famous elephants) travelled through Iberia, the south of France, and over the Alps in his famous crossing. The Romans, expecting a naval invasion, had been preparing for an invasion of Africa in Sicily. When news reached a paniced Rome that Hannibal was entering Italy, an army was sent north commanded by T. Sempronius Longus and P. Cornelius Scipio.

Hannibal defeated this army at the Battle of Ticinus, then finishing it off at the Battle of the River Trebia. The next year, Rome sent another consular legion (large army) under G. Flaminius, which was ambushed and devastated at the Battle of Transimene. Now, in 216, Rome sent both their executive generals (consuls) Varro and Paullus with an army numbering 86,000 against Hannibal's, which numbered roughly half of that.

The Romans lined up for battle and Hannibal did the same. He placed his weakest troops in the center, then as the army marched, the center marched way ahead of the flanks so the army looked like this from above: ^

Hannibal's weak troops in the center were the first to meet the Romans because of this and after some fighting they began to flee. The Romans followed and soon the massive Roman army was pouring into the gap, following Hannibal's center. The Romans lost their organization, which was the greatest advantage of their "manipular legion", which revolved around cohesion of troops in disciplined groups which could be manuevered before and during battle. Now, the Roman army resembled a horde of soldiers rather than an organized or disciplined army. Hannibal had kept his crack Libyan spearmen as reserves and sent them to head off the Romans, while Hannibal's flanks surrounded the Romans on all other sides. What followed was the greatest disaster of Republican Roman history. Livy says that 50,000 of the Romans were slaughtered (with another 20,000 being captured) while Polybius claims only 5,700 of the Carthaginians were killed. Paullus was killed, however Varro was able to escape with a large portion of the survivors.

After three massive victories, Hannibal had secured himself as one of the greatest generals in classical history, Cannae being his greatest moment. Although he would go on to lose the war, Hannibal would forever remain legendary in the eyes of the Romans he had nearly defeated.

Sources:

Adrian Goldsworthy's "The Punic Wars".

Polybius' "Histories".

In terms of military importance, it was a double envelopment. The wikipedia article has it under 'pincer movement'.

http://en.wikipedia.org/wiki/Pincer_movement

Unless the encircled troops can breakout, this ensures the complete destruction of the army, whether it be a mass surrender or wholesale slaughter.