Incredibly.

The end of the First Punic War (which itself lasted almost 20 years) in 241 BCE meant new territory for Rome: Corsica, Sardinia, and Sicily, the three major islands in between Rome and Carthage (Rome also got a bunch of smaller islands, but whatevs). Before the First Punic War, Rome owned just continental Italy, and the very tip wasn't even owned by Rome. For example, this was what the territory map looked like right before the First Punic War.

Carthage was the largest nation in the area and Rome could never have expanded so far without dealing with Carthage at some point. If Rome had lost the First Punic War, that would've crippled the nation (at that point about 500 years old) and it probably wouldn't've been as legendary as it was.

Furthermore, the First Punic War necessitated that Rome build a fleet and take command of the sea. Rome was virtually all land-based before the war, and the creation of a navy allowed Rome to secure its island holdings, conquer overseas lands, and improve trade.

The Second Punic War (the one where Hannibal's elephants crossed the Alps; this war, too, lasted about 20 years) was also crucially important in the development and expansion of the Roman Republic. By the end of that war, Rome took East Spain (and independently advanced North and East a little bit). Had Carthage won the Second Punic War, it might mean curtains for everybody's favorite nation (Rome).

The final chapter in Carthage's defeat was the Third Punic War, which never really threatened Rome's dominance anyway. After that, Rome seized Mauritania (modern day Morocco) and controlled the Strait of Gibraltar. Rome burnt Carthage to the ground and took the lands that Carthage once ruled.

TL;DR: The Punic Wars were the most important wars for the expansion of the Roman Republic.