This is a fairly broad question and covers about 100 years of history where a lot happened, but I will try to give both an overall view and specific information about each particular iteration of conflict.

Generally speaking, Carthage and the Roman Republic were competing for the same things. They were each seeking trade, influence, and resources in and around the central and western Mediterranean. This naturally put them at odds with each other. The three wars were separated by significant periods of peace and each had separate specific causes, but the underlying factors in each were the same.

The First Punic War, from 264-241 BC, was fought over control of Sicily. Rome had recently completed cementing hegemony over the Italian peninsula via their system of alliances that lasted through to the Social War in the mid-first century BC when the Italian allies were finally granted full citizenship status. Sicily is, of course, right next to Italy and a natural target for Roman expansion. It had been settled in centuries past by Greek colonists, and was home to several city-states that were culturally Greek. It began as a proxy-war, where two of the Greek city-states (Messina and Syracuse) become embroiled in conflict. Carthage sided with Syracuse, and Rome with Messina. What followed was a mostly naval conflict that ended in near total Roman victory. Carthage ceded any influence over Sicily and paid a large indemnity to Rome.

The Second Punic War, from 218-201 BC, is the most memorable of the three because it involved Hannibal's famed invasion of Italy. Again, it's direct causes belli was conflict between city-state proxies, this time an Iberian city that had been allied with Rome was attacked by Hannibal's army. Hannibal followed up by launching his invasion of Italy and won numerous great battlefield victories, but was unable to translate them into political victories. Eventually Rome recovered, counterattacked, and forced Carthage to capitulate. This essentially ended Carthage's status as a power in the ancient Mediterranean.

The Third (and last) Punic War, from 149-146 BC, resulted in the destruction of the city of Carthage. 50 years after the end of the last war, Carthage found itself on the receiving end of numerous raids from Numidian forces who were clients and allies of Rome. Carthage attempted to fight back, lost, and found itself again at war with Rome. They lost, and their city was burned to the ground. The city was then rebuilt as a Roman settlement, and persisted until the Arab conquest in the late 7th century AD.

My source for the Punic Wars is primarily "The Punic Wars: Rome, Carthage, and the Struggle for the Mediterranean" by Nigel Bagnall.