These were a series of wars fought between The Roman Republic and Carthage. The basic explanation for them seems to be that there were only two major powers left (though many minor ones), none to provide a third to assure Balance of Power and the Mediterranean sea just wasn't big enough for both.
The first one started when a group of mercenaries around what is now Messina in Sicily called the Mamertines declared themselves an independent kingdom and asked for protection from Syracuse. Their critical strategic position caused both Rome and Carthage to rush to offer their protection and to collide with one another touching off a general war. The Romans succeeded on land making themselves the dominant power in Sicily. At sea there was a remarkable upset in which the lubberly Romans took on the Carthaginians who were recognized as a Badass Navy and swept them from the sea (mostly by getting their own Badass Army from their ships to those of their enemies via a drawbridge-like device known as the corvus).
The Second War started in Spain in a similar manner, as a quarrel over the fate of a small state, this time the city-state of Saguntum. However it appears to have been a deliberately engineered provocation by the famous Hannibal Barca, a Carthaginian noble who was ruling Spain in the name of the Carthaginian empire but almost as a private kingdom. Upon the declaration of war, Hannibal made a surprise march over the Alps and invaded Italy, winning several battles including Cannae, a battle which is still a catchword among connoisseurs of tactical virtuosity. The Romans recovered by their desperate gathering from any possible source of Reserves including slaves. Deciding that they couldn't match Hannibal they pinned him in Italy using the strategy devised by Fabius Maximus Cunctator (Fabius the Great Delayer). In the meantime Roman legions began tearing up the rest of Carthage's empire to leave Hannibal isolated. During this time the general Publius Cornelius Scipio Africanus better remembered as just Scipio Africanus (literally Scipio [conqueror] of Africa), from the Scipio family gained prominence with his victories in Spain. On his suggestion the Romans landed close to Carthage forcing Hannibal to return to defend it. Scipio and Hannibal met on the field of Zama, and Scipio won leaving Rome as the greatest power in the Mediterranean and well on its way to becoming The Empire.
While the First War seems to have been a mutual collision, and the Romans could arguably claim to be fighting in self-defense in the Second, the Third War was a genuine Kick the DogWhile It's Down on Rome's part. The best that can be said in extenuation is that every family in Rome was mourning someone and they would be more than human not to want Carthage destroyed. Yet by that time, the Empire of Carthage had been reduced to the City of Carthage. The Third Punic War was thus simply the siege of Carthage, which didn't stand a chance from the start. The Romans overcame desperate resistance, sacked Carthage and according to legend "sowed the fields with salt" to keep anything from growing. This was the end of the wars.
Rome seems to have triumphed through its superior organization. It is also credited with a more motivated populace, as the army of Carthage was composed chiefly of vassals and mercenaries whereas the Roman army was made up of citizens and near citizens. Rome could also, for that reason, field a greater supply of manpower. Finally Carthage itself may have been lax in its support until too late, and the Second War at times looks like a private war of Hannibal's. The war created a number of famous commanders, mostly on the Roman side, but the best remembered was the Carthaginian Hannibal.
The term Punic Wars, by the way, comes from the term Punici or Poenici, which is the Latin word for Phoenicians (members of that civilization founded Carthage in the 9th century BC). Carthaginians may have called it "the Roman wars" for all we know, but their perspective is unfortunately lacking for obvious reasons. Most sources for the history of the Punic Wars are either Roman or Greek.
Also technically the longest war in history, due to the mayors of Rome and Carthage signing a peace treaty in 1985 "officially" ending the war.
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Absent-Minded Professor: Archimedes. According to tales, when the Romans were doing their Rape, Pillage, and Burn in Syracuse, a legionary came upon him while he was doing math problems in the dust. Archimedes barely bothered to notice him so the soldier killed him. Marcus Claudius Marcellus, the Roman General, was greatly annoyed as he had wanted Archimedes alive, either because the Roman general was a Cultured Warrior or Archimedes might be useful (Given that Marcus had been awarded the Spolia Optima, and was regarded before Scipio Africanus as Rome's foremost General, it's very likely to have been the former.)
The Alliance: Rome's strength was that it led a large one that provided huge allied contingents and defended it. Hannibal's strategic requirement before being able to directly attack Rome was to break it, and it was precisely his failure to get the Etruscan, Latins and Umbrian (the largest allies and the one directly surrounding Rome) to switch side that ultimately caused his defeat.
Invoked by Hamilcar, Hannibal's father: realizing Rome's strength and being Genre Savvy, he went in Spain to create one for Carthage, also including the Numidians for their cavalry. Differently from Rome's, Carthage's alliance fell apart: the Spanish tribes were either conquered or convinced to switch sides by Scipio, and the Numidians changed sides after Scipio inflicted them a Curb-Stomp Battle rivaling Cannae (using the survivors of Cannae to boot) and replaced their king with a Roman-friendly one in the wake of this victory.
Rome and Carthage were this from the Phyrric Wars until the First Punic War.
Ambadassador: Scipio was about as good at diplomacy as in war and the two reinforced each other.
Badass Army: Most obviously the Roman legionaries, but Hannibal's army in Italy qualifies, what with Hannibal managing to out-strategize Rome for 16 years after losing almost half his army in the Alps.
Badass Creed: The Mamertines, whose name means Children of Mars (the war god). In other words they were the "Children of battle".
Badass Family: The Barcids (Hannibals family). Several families in Rome including the Scipios, of whom Scipio Africanus was best remembered.
Boarding Party: The Romans stopped losing sea battles once they stopped trying to use traditional tactics (which didn't work, since the Carthaginians were better at it) and started using these, relying on their infamously brutal legionaries to serve as ad hoc marines.
Boring but Practical: Fabius' "war of attrition" tactics. Didn't win him any friends, was way more effective than than just just charging at Hannibal. Figuring out a way around the Fabian strategy was something the Carthaginians did by the skin of their teeth, and mainly consisted of "trick Fabius' men into attacking oxen," because by that point Hannibal was surrounded.
Cincinnatus: Fabius and Scipio. Scipio could conceivably have made himself a Glorious Leader. Fabius was a "dictator" (in the Roman sense it was a legitimate office appointed by the Senate in emergency in a decree of what moderns would call martial law) for awhile and left dutifully when his time was up.
Curb-Stomp Battle: The infamous Battle of Cannae may be the single biggest curb stomp battle in recorded history. Using a vastly outnumbered army consisting of warriors from different lands, Hannibal managed to lure the tightly packed, phalanx-like formation of the Roman army into his deliberately weakened center. The Carthaginian army bowed backward as the Romans pushed in. Once the Roman army had completely waded into Hannibal's odd formation, he sent his cavalry around the back to attack the rear, completely encircling the Romans. 50,000 Roman troops died; Carthage lost 8,000.
Defeat Means Friendship: After the First War; Syracuse and Rome. During the second war the Romans weren't so merciful.
Determinator: Rome, as best shown by when its forces were defeated at Cannae: when asked if they wanted to have another go at defeating Hannibal, the survivors followed Scipio Africanus in Africa and actually defeated Hannibal.
Hannibal became this for the Romans after Cannae, to the point that some young Romans proposed to leave the city and found a new one elsewhere. It mostly subsided when the Romans realized Hannibal couldn't hope to win a siege or even come close to Rome, but they still feared him in open battle;
Marcellus earned this fame in the Sicilian Campaign by being so ferocious with the local cities that even the Senate (who usually didn't give a shit to what happened to rebels and traitorous allies like those that were dealing with Marcellus) considered him going overboard. At the start they just hated him, with even neutral and Roman-aligned cities declaring allegiance to Hannibal, as the bulk of his army was busy besieging the unconquerable Syracuse and all they had to deal with were raiding parties, but they started fearing him when the unconquerable Syracuse was stormed andsacked, to the point they immediately switched sides. And when Marcellus was elected governor of Sicily, the Sicilians sent envoys to beg and bribe the Senate to sent him somewhere else before he could even start the travel (the Senate, given what had happened in the past, sent him to Apulia, where he decided to go easy on the locals).
Egopolis: The Spanish city of Barcelona was named after the Barcid family.
Enemy Mine: They didn't like it, but when the Mamertines requested for an alliance with the Roman Republic, the Syracusans allied with their former enemy the Carthaginian Empire.
Even Better Sequel: The Second Punic War. It's the one people pay the most attention to, and with good reason:
Hannibal crossing the Alps.
The Battle of Cannae.
Scipio Africanus invading Spain.
Marcus Claudius Marcellus obtaining the spoila opima — claiming the armor of an opposing commander after killing him in single combat.
The Battle of Zama.
Even Evil Has Standards: Roman treatment of cities that were either in rebellion or traitorous allies was basically genocidal, but Marcus Claudius Marcellus went so far with the Sicilians that even the Senate was forced to admit he had gone overboard.
Folkhero: Several. Hannibal was a heroic figure for the Carthaginians, the Macedonians and other enemies of the Romans (and even the Romans had some level of admiration for him). Scipio and Fabius Maximus for the Romans. Hieron II for the Sicilians and other Italian peoples.
Marcellus also gets bonus point for earning himself the Spolia Opima (lit. Ultimate Spoils), an honoreficence bestowed only to a general who personally slays the enemy commander during a battle (an event so rare that, including Marcellus, the spolia opima was awarded to only three persons, with the first Romulus. That's right, to earn that you had to become a living legend). He did it before the Second Punic War, when, at the Battle of Clastidium against the Gauls, he spotted a Gaul with a cool armour and killed him so fast he didn't find out it was the enemy general until the Gauls saw what he had just done and started running.
Galley Slave: Averted. Galleys on both sides were usually rowed by freemen.
Guile Hero: Scipio is usually portrayed as either this or a Magnificent Bastard. Hiero II of Syracuse was definitely one cunning bastard. The whole First Punic War was a Batman Gambit he pulled on the Mamertines, and he played both Carthage and Rome against them. First by allying with the Carthaginians, then convincing the Romans after they won that he didn't had any real grudge against them, that he just wanted the Mamertines and that he would be a much better ally than them. Syracuse became a small superpower thanks to his diplomacy.
Honor Before Reason: The Roman general Regulus was captured by Carthage, but while a prisoner, he was sent back on the promise that he would advocate a peace favorable to Carthage. When he arrived in Rome, he urged the Senate to refuse any peace offers and continue fighting. Afterwards he voluntarily returned to captivity with the now irate Carthaginians, but according to legend, Regulus submitted to death by Cold-Blooded Torture rather than break his parole.
Hope Spot: Carthaginian commander Hasdrubal leaving the Iberian Peninsula with a large army and invading Italy late into the Second Punic War to join forces with Hannibal was this to the latter. Hasdrubal ended up defeated and killed by Romans in the battle of Metaurus River, and the Roman commander Gaius Claudius Nero had his head thrown into Hannibal's camp.
Hopeless War: The Second War when it became clear that Rome had a larger resource base, that its alliances with the Italian cities were stronger than Hannibal hoped, and that Fabius would refuse to challenge Hannibal on his terms. Rome won because it was always strategically dominant in the big picture. The Third War was never going to end well for the Carthaginians.
Improvised Weapon: During the final siege, Carthaginians made bowstrings out of their women's hair.
Magnificent Bastard: Romans seem to have considered Hannibal this, which is why we still remember him today. They hated him too much to regard him as a Worthy Opponent, but they regarded him as scary and formidable.
Manly Tears: It has been told that Scipio Aemilianus, the commander of the Roman army in the Siege of Carthage, cried his eyes out when he withnessed the destruction, the slaughter and the looting his troops commited after they breached through the Carthaginians' defences.
Although it is worth mentioning that it was partly out of horror, but also because Scipio Aemilianus mused on the mortality of all nations. He basically thought "this will happen to Rome one day". Fast forward roughly 600 years, and you see that it does happen. Thrice.
Officer and a Gentleman: Scipio. He became an exemplar of Roman virtue for generations afterward. One story tells of him, having captured Cartagena, being brought a beautiful Spanish noblewoman as a prize by his men, who knew he had a weakness for beautiful woman. Instead of keeping her as a Sex Slave, he located her fiance and reunited the couple. The man turned out to be an important Iberian chieftain who promptly joined the Roman cause.
Overly Long Gag: Famously, Cato the Elder often invoked his personal motto "Carthage must be destroyed," even in completely unrelated contexts. He continued to do this for many years until he died (shortly before the Third Punic War broke out).
The Third War qualifies in another way... a peace treaty was officially signed in 1985
Paper Tiger: Amazingly enough, Hannibal's Badass Army was this, being very small and reinforced by allies of dubious loyalty, and completely useless in a siege or in any battle he couldn't outmanouver the Romans. Hannibal's entire strategy was centered on hiding this weakness until enough of Rome's allies switched sides or the Roman morale broke down and they sued for peace. It nearly came to fruition after Cannae, but Fabius Maximus managed to stop the morale fall, and when Hannibal decided to offer them a chance to surrender at decent terms the Romans realized this, raised a bigger army... And demanded the rent for the public land occupied by his camp.
Patriotic Fervor: Romans had a lot of this. Carthage is said to have not so much.
His dad, Hamilcar Barca, hired a lot of mercenaries after the First Punic War and basically made a private army. Over time, they changed from swords-for-hire to loyal followers of the Barca family. A fair number of these men served Hannibal during the Second Punic War, and even followed him in the gruelling march across the Alps.
The Romans were rightly infamous for systematically killing all the elders and nobles and enslaving everyone else upon storming a city, and adding all able-bodied men to the murdered people if the city was in rebellion or a former ally. This was crucial in the Sicilian campaign: upon realizing that not even the well-defended Syracuse could resist a Roman siege and that the Romans in Sicily were the supposedly cowards that had escaped Cannae, the other Sicilian cities begged for peace before the Romans could spare supposedly stronger troops. It helped that the Roman commander Marcellus had spent most of the Siege of Syracuse leading raiding parties to raze villages with such ferocity that even the Senate considered him going overboard.
Red Baron: Fabius, Marcellus and Scipio: Fabius was known as Cunctator ("delayer") and Shield of Rome for his guerilla tactics that screwed Hannibal's strategy and nearly annihilated him, Marcellus was known as the Sword of Rome for stalemating Hannibal every single time they fought and crushing any and all of Hannibal's allies who had the bad sense of not surrendering as soon as he showed up, and Scipio's nickname Africanus, while literally translated as "The African", was Roman slang for "he who went to Africa and kicked the ass of anyone who dared to not worship Rome, including freakin' Hannibal".
Screwed by the Network: Hannibal was unable to follow up his stunning victory at Cannae with an assault on Rome itself partly because he received very little support from Carthage. It's possible that this was because political opponents there, not wanting him to achieve even greater glory, withheld aid.
Silly Reason for War: A small local conflict between a Greek city and a small horde of unemployed mercenaries escalated into three wars that were the ancient history's versions of World War I, II and III.
Single-Issue Wonk: Cato the Elder. That quote at the top? He used it to finish every speech, regardless of subject.
Storming the Castle: Scipio's attack on Cartagena. He actually did it marching across the harbor, after learning from intelligence about tidal quirks in the area that allowed that in places.
Trope Namer: "Fabian tactics", so called after Fabius Maximus Cunctator.
Troll: Rome before the start of the Third Punic War, though they were more than likely trying to provoke Carthage. After Carthage raised an army without its permission in order to try and drive off marauding Numidians, Rome declared that it would need to "appease the Roman people", by giving 300 children from high-up families to be raised in Rome. Then hand over all of their armor and weapons. Then move ten miles inland and burn the city.
Victory By Endurance: The Fabian Strategy in a nutshell. The Roman leadership, for the most part, knew that the strategic situation meant that Rome had time on her side. All they had to do was slowly grind Hannibal and the Carthaginians in a war of attrition, rendering Hannibal's superior tactics useless.
We Have Reserves: Rome's army was essentially a citizen militia, supplemented by troops from its vassals allies. This gave Rome a significant manpower advantage over Carthage, which relied on mercenaries. The decisive factor in the wars was Rome being able to replace its losses, which were at times quite staggering.
In the First Punic War, the Romans lost almost their entire fleet in a storm off Sicily in 255 BC (280 ships, about 100,000 men). They built and manned a new one, which perished in another storm en route to Africa in 253. Then they lost a third fleet in another storm off Sicily in 249. Finally rich merchants, citizens and shipowners built a fourth fleet, which pulled off the decisive victory of the Aegates Islands in 241.
Worthy Opponent: Seven years after Zama, Hannibal was in exile at Ephesus, where he was honorably received by Antiochus III of Syria, who was preparing for war with Rome. At one stage, Scipio Africanus came to Ephesus with some others to hold talks with Antiochus, but he was elsewhere, so Scipio and his colleagues had to wait. An amusing, but likely apocryphal anecdote recounted in several histories has Hannibal and Scipio meeting here, and amicably discussing the greatest military commanders of all time. Hannibal and Scipio were, of course, both able to agree on the greatest - Alexander the Great. Scipio then asked Hannibal who he considered the second greatest. It seems Scipio expected himself to be named second greatest, as he was quite nettled when Hannibal gave his answer: Pyrrhus of Epirus - he of thePyrrhic Victory. So he asked who the third greatest was, expecting that that title, at least, could be awarded to noble Scipio. But it was not to be; Hannibal named himself the third greatest, and gave several reasons why. Scipio, seeing that Hannibal was likely to go on for a while, laughed, asking Hannibal where he would rank himself if he had not been defeated by Scipio. Hannibal replied "In that case I should have put myself before Alexander."
Also, Hannibal considered Marcellus as this, due the Roman general having stopped his advance five times and single-handedly ruining Hannibal's strategy to try and break Rome's Federation by personally retaking or destroying a number of key cities that had defected and scaring into surrender the entire Sicily (the latter stopped Carthage's attempts at supporting Hannibal, as they now had no foothold from which to send troops). It's said that, upon his death when a recon patrol he was personally leading was ambushed by Numidian cavalry, Hannibal gave Marcellus a military funeral before sending his ashes back to his son.
Xanatos Gambit: Scipio's invasion of Africa, once the Numidians switched sides: whatever happened, Rome had won the war. Carthage was conquered before Hannibal could be recalled? Hannibal and his army would have suffered a devastating morale blow and found themselves with no way to receive meaningful reinforcements, making easier to finally destroy them. Hannibal was recalled? He had to go by ship and the Roman navy ruled the Western Mediterranean, meaning there was a good chance Hannibal and his men would get sunk during the voyage. Hannibal returned to Africa and defeated Scipio? He still had to deal with the Numidians, and would have no chance to invade Italy again, especially as the Romans would finish his allies in Italy. In fact, Scipio's victory at Zama just served to make the Carthaginian Senate realize they had lost (Hannibal already knew and was just following orders and trying to earn a favorable treaty).
Hannibal: Rome's Worst Nightmare is a made-for-TV movie produced by the BBC that retells the story of Hannibal with refreshing accuracy. Not exactly fiction, but worth mentioning.
Ross Leckie's Carthage Trilogy retells the story of the Second and Third Punic wars from Hannibal's perspective.
In Poul Anderson's Time Patrol story "Delenda Est", tampering with the Punic Wars produced an Alternate History and the Time Patrol must straighten out the battle that the Scipios should have survived to put it back.
In Vampire: The Masquerade, the Punic Wars served as a backdrop to and was to some extent engineered by two warring clans of vampires, the Brujah in Carthage and the Ventrue in Rome.
Rome: Total War and Total War Rome II allow you to reenact the wars in their Grand Campaigns and have several famous battles as separate scenarios. The latter also has a DLC campaign called Hannibal at the Gates centered around the Second Punic War.