They opened by setting the scene - Rome and Carthage each with an empire facing off across the Mediterranean. The Carthaginian empire had grown from Phoenician trading outposts, and was centred on the western Mediterranean with important holdings in Sicily and Spain as well as their North African heartland. Rome, obviously centred on Italy, were the new kids in town and the first war between Rome & Carthage (the First Punic War) started over control of Sicily. Hannibal's father Hamilcar Barca was the general in charge of the Carthaginian army during the last part of this war - he didn't win, but it seems he was probably never actually expected to win as the army was underprovisioned for that sort of undertaking. Also after the end of the war (Carthage lost) he wasn't punished, which supports this - the Carthaginians had no qualms about executing generals that they felt had failed.
After the First Punic War Carthage's finances were in trouble, which was even more of a problem than you might expect because their army was made up of mercenaries who promptly revolted when they weren't paid. Hamilcar put down this rebellion, and then took the army off to Spain to secure & improve the Spanish holdings - Spain being a major source of wealth for Carthage due to the silver mines. And this is where Hannibal enters the story - he's 9 at this point, and gets taken along with his father to Spain to live and campaign with the army. And this is really where & how he learns to be such a good general - he lives with the army so knows the men and what will motivate them etc. And gets to see how to run campaigns first hand.
Once Hamilcar dies (in battle) Hannibal's brother-in-law takes over the army and gives Hannibal the job of cavalry commander. Once his brother-in-law dies (assassinated) Hannibal gets the job of overall commander - he's only about 25 at this point, but has 16 years of military experience. Then Second Punic War kicks off - starting over control of a particular Spanish settlement which is under the protection of Rome. And right from the outset Hannibal demonstrates some of the genius for which he's remembered - one of his key qualities is the speed at which he (and his army) reacts to events. Rome warned Hannibal off attacking the Spanish settlement, but by the time either Rome or Carthage had reacted Hannibal had already seiged and razed the city.
This is the point where Hannibal starts the journey that he's most remembered for - he marches his 80,000 strong army north through Spain and into Gaul, then east to the Alps and across them to northern Italy. He had to do this because Carthage were no longer the major naval power in the Mediterranean, and if they'd sailed to Rome they would have had even more trouble. They had to fight their way through both Spain and Gaul, and provision the army from the land they march through. Then they crossed the Alps during the winter. This whole journey reduces his army to approximately 26,000 men - through desertions, through leaving garrisons behind en route and through deaths as they cross the Alps. He still has some of the war elephants too! Despite these crippling loses Hannibal goes on to win over 20 battles against Roman armies that outnumber him, due to his superior tactical skills (including paying a lot of attention to and making use of the lay of the land he's fighting on) and the superior mobility of his troops. At one point he has the chance to march on Rome, but doesn't take this opportunity - opinion is divided on whether this was the right decision. Perhaps he might've won the war if he'd marched on Rome then, perhaps he was right and didn't have enough troops to properly seige the city for long enough for it to fall.
Eventually the Romans stopped trying to meet his army on his terms - instead they used delaying tactics to avoid battle but keep the army occupied. And then attacked Carthage itself forcing Hannibal to bring his army back home to save the city. This is the battle he lost, losing the war for Carthage - a while afterwards this was used as an excuse to send him into exile and he lived in a series of provinces/kingdoms that were being threatened by Rome offering them his advice on how to campaign against the Romans. Eventually one of these handed him over as part of their peace treaty with Rome, and rather than be captured he took poison.
Hannibal is still remembered (and respected) today because the Romans were impressed with him, and afraid of him. He was apparently used as a bogeyman for Roman children. And his tactical skill was respected even into modern times.