Rome vs. Carthage. There were actually 3 Punic wars though they were somewhat related to each other.

To set the scene, its the mid 200's BC. Rome owned pretty much all of Italy, and Carthage owned the north coast of Africa, southern Spain, and most importantly, a large chunk of Sicily. I'm going to give a very brief overview, feel free to ask for further details.

First Punic War: The stage is Syracuse, the biggest city in Sicily, an independent state not allied to either Carthage or Rome. The leader Agathocles hires a huge amount of mercenaries (called the Mamertines) from Italy to wage a war against Carthage. He loses. Fast forward 15 years or so, Agathocles dies, leaving behind a huge force of mercenaries that are no longer on anyone's payroll. They do what bands of warriors tend to do when left unsupervised, and start ravaging and pillaging the independent areas of Sicily. Eventually, the city of Syracuse has enough, and takes the fight to the Mamertines. The Mamertines appeal to both Carthage and Rome for help at the same time. Carthage responds and joins the war against Syracuse. However remember, Sicily is right off Italy's coast. If Carthage beats Syracuse, it would pretty much complete Carthaginian ownership of Sicily. As you can imagine, Rome doesn't want any potential enemies to have such a big power base right next to their home. So, hoping to avoid this, and take Syracuse for themselves, Rome gets involved too and the two states begin to battle.

The conflict escalates, and to cut a very long story short, the fighting spreads to cities all over Sicily, and eventually Rome builds a navy, tries (and fails) to invade North Africa, but eventually beats Carthage by utterly smashing their navy at the battle of the Aegeates Islands. Without naval support, the Carthaginian commanders on Sicily are stranded and without backup so are forced to surrender to Rome.

This sets the stage for the conflict between Rome and Carthage in the Second Punic War; the famous one with Hannibal and Cannae and whatnot. Essentially, until this point Rome had only been fighting battles in Italy, and therefore all its enemies were fairly close to home. Rome had adopted an attitude whereby, once an enemy was defeated, it had to be utterly destroyed or completely absorbed into their own "empire". There was no sense to them behind making peace and going back to the way things were. However, Carthage wasn't really destroyed by any means, though it lost the war. Carthage was also too far away from Rome for them to totally conquer and dominate them. What this led to was Rome treating Carthage as they usually did defeated enemies - by acting as if Rome had crushed them and Carthage had to accept whatever terms given. Carthage did not see it this way, as far as they were concerned they still had a vast power base, and had only lost a few overseas battles (and not necessarily even by that wide a margin). They felt the terms of their surrender were far too harsh and not really reflective of the situation. This dynamic continued for a long while and as I'm sure you can imagine, really deepened the enmity and hatred between the two states, setting up the ground for the Second Punic War.

I'm going to leave it there, since you asked simply what the wars were about and I think I've conveyed the root causes behind the wars, and you didn't ask for the whole story. I can certainly elaborate/expand if you want to hear more.

The short answer is that the Romans had built up control over the Italian Peninsula while the Carthaginians were establishing hegemony in Africa and the Western Mediterranean. Their spheres of influence came into conflict --especially where they had allies on the opposite sides of small troubles-- and from incidents that really should not have warranted total war things rapidly spiraled out of control in a way neither side expected. The first war saw Rome eventually strip Carthage of much of the middle ground over which the two powers had been jockeying for control. Carthage rebuilt its strength in Spain to compensate for what it had lost, and the second war was truly a life and death struggle between the two foes. Carthage eventually lost --but not for lack of trying-- and the third war was basically Rome smothering Carthage to make sure it would never mend itself again.

EDIT: things, not thinks.