The first Punic War was basically the early Roman empire deciding that the Carthaginian empire's expansion into Sicily was too close for comfort. Much like modern day proxy wars the first Punic war started when two cities got into a scrap: one backed by the navy of Carthage and one backed by Rome. Rome wins once they figure out how to make legions work well on the ocean via platforms that hooked onto enemy ships. There is a lot more to it but the naval disparity during the war was a critical point.

Between the first and second war Rome manages to grab a few other islands from the control of Carthage and the Barcas of Carthage build up an empire in Spain. Coupled with a few other events the hard feelings propelled them to war.

This is when you get Hannibal Crossing the Alps and his great victories like Cannae. Unfortunately, Carthage was in no position to assist him in taking advantage of his victories so he was forced to constantly be on the move to feed his army and keep roman client cities from providing assistance.

Meanwhile Scipio, who had passed Hannibal as he went to Italy but was told by the Senate that they could handle it, was actually dismantling the Barca empire in Spain (lead by Hannibal's uncle). Once he completed that he sailed to north Africa to take the fight to Carthage. He did very well essentially causing the carthaginians to panic and recall Hannibal from Italy for a final battle.

But they supplied him with poor troops and he had to abandon his hardened fighters in Italy. The fight went poorly for Hannibal, Scipio won and was awarded the title Scipio Africanus by the Senate.

Carthage agreed to a humiliating treaty with heavy reparations, releasing their client tribes of Numidia and returning land to them, and demilitarizing themselves.

Between the second and third war a senator named Cato the elder finished all his speeches with "Carthago delenda est" or "Carthage must be destroyed." Carthage was recovering quickly from the second war. They had paid their reparations early. Fear of another bloody war lead the Romans to take action.

They sent legions over under the control of Scipio, a different one unrelated to Africanus, though it is a bit of a spoiler to tell you he also go the title Africanus. The large force panicked the elders of Carthage who met with him. Scipio demanded that they turn over all the weapons in all the cities.

Carthage had been significantly cowed by two bloody wars and they agreed and turned over ALL the weapons. Then they demanded that Carthage agree to permanent Roman sovereignty essentially making them a part of the Roman empire. That was too far they said no.

War were declared.

But cities are hard to take especially at that time. The various cities around Carthage were taken fairly quickly but Carthage held out for years. When the walls were finally breeched it was bloody urban fighting with everybody in the city armed to the teeth. Carthage was destroyed and the land returned to the Numidians who became a client state.

Source: Adrian Goldsworthy's Punic Wars which I highly recommend for the casual reader.

Different people give different reasons. Bagnall said P1 was just war fever on the Romans' side. Polybius said it was because Rome feared Carthaginian encroachment on Italy if they took Messina. Cassius Dio said both sides really wanted a fight and Messina was just an excuse.

P2 according to Livy, was entirely due to Hannibal's inherited hatred of Rome. The casus belli there was the siege of Saguntum, and it's not clear to me who was in the right on that. The treaty Rome signed with Carthage after P1 was something to the effect that the river Ebro in Spain delineated the two separate spheres of influence, and that Carthage would not attack Roman allies. When Hannibal destroyed Saguntum, the Romans complained about a breach of the treaty, and Carthage replied that Rome had no business having allies south of the Ebro. It's weird that Rome didn't intervene until Saguntum had been destroyed, even though they knew it was besieged. Some people speculate that this was deliberate to get a casus belli.

P3 was Rome's intervention in a border dispute between Carthage and Numidia, who were both technically Rome's allies at that time. The treaty signed at the end of P2 included the provision that Carthage would not wage war without Rome's permission. When the Numidian king Masinissa encroached on Carthage's territory they responded with force. Rome took the Numidian side on that issue. RIP Carthage.

Why Rome chose Numidia is a question. It could be because of international politics, but it could also be about internal politics - advancing the careers of Cato and Scipio Aemilianus. In the latter case there would have been war either way. 146 BCE is a strange year in Roman history. Romans weren't in the habit of destroying cities, but that year they destroyed both Carthage and Corinth, and they didn't do anything on that level until Jerusalem in 70 CE.

Well it was mostly fought, as with most wars, over trade and territorial rights. Carthage had become prosperous despite owning very little territory directly (they had many colonies but they maintained hegemony over them more than direct rule) by trade and dominated the Western Mediterranean. Obviously, Rome wanted a piece of that trade as well but the most valuable ports were in Sicily, which was mostly under the thrall of Carthage. To add to that, Rome always viewed Sicily as its territorial right, as the Romans saw themselves as the rulers of all of Italia. From there, a complicated mess involving some mercenaries sacking a city and said city calling both Rome and Carthage to help ended up dragging the two into war. A war which Rome won.

The second war began when the vengeful general Hannibal decided to attack a Roman ally for a supposed breach of a treaty. Hannibal's family, the Barcas, had built an empire in Spain that grew rich off of gold and silver mines, so much so that they could afford to pay back all the reparations owed from the first war and hire a new mercenary army. Hannibal's plan was based around riling up the socii (tributary states of Rome) into open revolt against the Romans since Carthage lacked the logistical capacity to aid him. Despite early victories, Hannibal underestimated the Roman's resolve and warlike nature, and although he was a good tactician he lacked strategic know-how. While he piddled around Italy for 15 years beating up armies of raw recruits, Rome's hardened veterans dismantled his empire in Spain and killed his brother, Hasdrubal. They eventually went for Africa himself and Hannibal was defeated for the first time.

The third war was almost a matter of closure by that point. Carthage had basically been reduced to the status of city-state and Rome actively dominated her affairs even in peace. They were no longer very rich or powerful, but the Romans now had ambitions for empire-building and the first challenge was to finish what they started with Carthage's conquest. The city was taken and sacked, but it remained incredibly prosperous and important in the Roman Empire. Much of modern day Tunis is built on the ruins of Carthage, and some areas of the ancient city can still be viewed in the oldest quarters of Tunis, albeit there isn't much left.