It was a rivalry between up and coming Rome vs established Carthage (city itself was in present day Tunisia, but they controlled a lot of northern Africa and Iberia). that turned violent regarding the ownership of Sicily. Carthage started out with the advantage but lacked the overall strategy to win and eventually backed down when Rome built a navy from scratch and then iterated a few times to get a design they couldn't deal with.
But then a Carthagian general named Hannibal wanted another go and this time snuck an army, including elephants, over the alps and then spent a decade ravaging Italy while Rome couldn't really do anything about it, despite Carthage refusing to send any reinforcements. Well, eventually Rome realized that throwing legion after legion at his army wasn't working and instead threw some legions at Carthage itself. They also converted some Carthage allies that were tired of Carthage throwing its weight around. This led to Carthage recalling Hannibal and then losing a decisive battle, at which point they basically became a hostage state, needing to pay Rome a huge sum regularly.
Anyways, time passes and eventually some Roman senator visits Carthage and is appalled to see them still living better than Rome and goes back to convince Rome to go a 3rd and final time. They seige the city, at one point the lady of the city kills herself and her children to keep them out of Roman hands, which did prevent them from being unwilling participants in the genocide that followed, which included the city being dismantled brick by brick and anyone who wasn't killed taken as a slave.
And that's basically the origin story of Rome as the regional power. Prior to that was another set of wars uniting the Italian boot under Rome. After that, they controlled all that Carthage land and then stated expanding into where France is today, leading to the rise of Julius Caesar, him seizing control of Rome by breaking an important rule with no teeth, and then his assassination (because he wasn't on the rich fuck side), followed by the first Roman emperor.