Carthage started out as a trading colony of the Phoenicians, folks who had a big shipping empire at the eastern end of the Mediterranean and who also claimed they invented the alphabet. By 500 BC there were Phoenicians trading posts all along the northern coast of Africa, through Gibraltar, up the Atlantic coast of Spain, and all the way to England. Phoenicia eventually was laid waste by the Persians, and Carthage and the other trading posts became orphan colonies. Centrally located Carthage quickly dominated the other former colonies and spread into inland and Mediterranean coastal Spain and the Mediterranean Islands, becoming the dominant local naval power in the process. Rome, meanwhile, was being founded and eventually became master of the Italian peninsula. When the two empires started to impinge on each other, sparks were sure to fly. The main bones of contention were initially Sicily and the Mediterranean coast of Spain. Historians differ on which side was the most treacherous and who started the decades-long conflict, but the outcome was clear -- Rome won all three Punic (i.e., Phoenician) Wars, and by about 145 BC Carthage disappeared as a military and commercial power (which was a good thing for Rome, because it soon had to deal with the Macedonians, onetime allies of Carthage.) The Romans acquired North Africa, and the need for a professional Roman army was established (and that led inexorably to the end of the Republic).
Internet Links:
The Punic Wars:
http://history.idbsu.edu/westciv/punicwar/index.html
http://ancienthistory.about.com/library/weekly/aa012798.htm
http://ancienthistory.about.com/library/weekly/aa020398.htm
http://ancienthistory.about.com/library/bl/bl_3d_punic.htm
http://ancienthistory.about.com/msub_punic_wars.htm
Hannibal
http://www.fordham.edu/halsall/ancient/polybius-hannibal.html
Cannae, Rome's worst ever battle day, Hannibal's best
http://www.fordham.edu/halsall/ancient/polybius-cannae.html
Cato and the end of Carthage
http://ancienthistory.miningco.com/education/ancienthistory/library/bl/bl_3d_punic.htm
PS--
1. According to some historians, only one elephant, named Abullah Bassan, actually made it through the Alps. The others supposedly froze or fell and became steaks and chops. Hannibal reportedly was able to ferry about two dozen more across from Spain.
2. The Carthaginians were as bad
as the Romans said -- lots of baby burning to eliminate bad karma, etc.
-- and they deserved to be destroyed.