"[The Carthaginian general Hannibal] was born in 247 BC, the eldest son of Hamilcar Barca. After making Hannibal swear an oath never to be a friend of Rome, Hamilcar took him to Spain in 237, where he stayed during the commands of both his father and his brother-in-law Hasdrubal (1), marrying a Spaniard from Castulo." Source: Oxford Classical Dictionary (OCD, 3rd ed. revised).
(1) The OCD describes Hasdrubal as Hannibal's brother-in-law while Wikipedia calls him Hannibal's brother. As it turns out, there were at least four different Hasdrubals in Spain at the time. One of them was Hamilcar's son-in-law; the other Hamilcar's son.
Rome's expansion brought it into conflict with Carthage, the other great Mediterranean power of the time. Both Hamilcar and Hasdrubal (Hasdrubal the Fair) were leaders during the First Punic War (264-241 BC). Apparently Hasdrubal [the brother-in-law] assumed the supreme command in Spain after Hamilcar's death and Hannibal took over after Hasdrubal died, reverting to his father's policy of attacking various tribes to the north. According to the OCD, Hannibal started the Second Punic War by attacking and eventually defeating a Roman city called Saguntum located on the eastern coast of Spain:
"Regarding Rome's alliance with Saguntum (Sagunto) as a threat to Carthage's position in Spain, he [Hannibal] decided to defy her, and put pressure on Saguntum. He rejected a Roman protest, and after consulting Carthage began the siege of Saguntum in spring 219, knowing that war with Rome would result, and took the city eight months later." (OCD)
Hannibal decided to invade Italy without bothering to wait for the Romans to declare war. His goal, apparently, was to weaken Rome by stripping away its allies, rather than to destroy it outright. According to the OCD, "he left his capital, Carthago Nova (modern Cartagena) in May 218, with a professional army of 90,000 infantry and 12,000 cavalry (Iberians, Libyans and Numidians) and [40?] elephants, leaving his brother Hasdrubal to hold Spain... [emphasis added]"
Hannibal's famous crossing of the Alps took around 15 days and no one is sure which route he took, though new evidence surfaced a few years ago which may clear up the matter. Carthage was eventually defeated at the Battle of Zama in 202 BC by the Roman general Scipio Africanus., aka Scipio the Great, "one of the greatest generals and military strategists of all time," according to Wikipedia and every other source I've read. An excellent overview of the complex series of Punic Wars can be found at the Ancient History Encyclopedia.
Related: Carthage - The Rise And Fall.