The answer, as always, depends on the period of time you're talking about. The Roman navy changed even more dramatically over the course of its existence than the army did.

The Republic

For the early republic, the answer is simple: there was none. Or not much of one. The first we hear of a Roman navy is in 311 BC, when two officials were appointed to command two squadrons of ships (probably triremes). These subsequently aren't mentioned in our sources, except in Appian who blames the start of the Tarantine War (and the subsequent Phyrric War) on a naval incident:

[The consul] Cornelius went sight-seeing along the coast of Magna Graecia with ten decked ships.^1 At Tarentum there was a demagogue named Philocharis, a man of obscene life, who was for that reason nicknamed Thais. He reminded the Tarentines of an old treaty by which the Romans had bound themselves not to sail beyond the promontory of Lacinium. By his passion he persuaded them to excitement against Cornelius, and they sunk four of his ships and seized one of them with all on board. — * Appian, the Samnite wars*

Not the most glorious start of Rome's naval history. (And one has to wonder: why did the consul go 'sight seeing' with half the Roman navy?) For the rest of this period, Rome mostly relied on allied cities with bigger fleets to provide them with ships.

This changed during the Punic Wars. Carthage was a naval power of unrivalled strength in the western Mediterranean, so Rome set out to rival them by engaging in a truly massive program of shipbuilding. Hundreds of large quinquereme ('with five oar banks'^2) warships were constructed, and almost as many were lost, either to enemy action or to flaws in their hasty construction^3, as storms devastated entire fleets. Eventually, the Romans came out on top. By the last battle, the Aegates islands in 241, the Romans were clearly equal if not superior to the Carthaginians in seamanship and skill, outmanoeuvring and devastating their enemy fleet with skilful ramming actions. (A few years back archaeologists actually discovered the location of this battle, which is very rare for ancient naval battles, and found several ship's rams as well as other assorted debris.)

In between the Punic Wars, the Romans used their new fleet to deal with the Illyrians, who were notorious pirates threatening the Adriatic, and thus started their expansion into the eastern Mediterranean. After the second Punic War, this proved the start of a new trend as Rome came into contact with the eastern Hellenistic kingdoms, such as Macedon, Egypt and the Seleucid Empire. These each maintained powerful navies, in part as a matter of prestige, with some truly massive warships such as hexērēs,^4 ('six') dekērēs ('ten') and even 'sixteens', though whether these were actually of any practical use remains doubtful.

In any case, in part because of their experience in the Punic Wars and in part because of their alliance with the powerful city of Rhodes and their highly skilled navy, the Romans quickly gained the upper hand at sea as well as at land, winning another series of naval victories against the Macedonians and Seleucids.

In the subsequent peace treaties, Rome forbade its rivals to maintain strong fleets. This had unexpected long-term-consequences, as Rome did not step up to fill the vacuum, and the eastern Mediterranean became rife with piracy in the 1st century B.C. And when I say piracy, you would not be far wrong in imagining "pirates of the Carribean" style shenanigans: the pirates formed entire fleets, kidnapped Roman officials from the coast of Italy, (and also a young Julius Caesar, at one point) raided the port city of Ostia just next to Rome, threatened the Roman grain supply and even defeated the Roman fleet in battle. One wonders, at this point, whether 'pirate' is even the appropriate term any more.

Eventully, Pompeius Magnus was appointed to deal with the threat and was given wide-ranging authority and a massive navy, which he used to quickly subdue the problem, (Leading some historians to wonder whether it was ever as big a problem as our sources make it out to be.) in considerable part thanks to large-scale resettlement projects where surrendering pirates would be given farmland away from the coast. This, however, set the stage for the role of the Roman fleets in the next round of civil wars: Caesar against Pompeius, the triumvirate of Octavianus, Marcus Antonius and Lepidus against Sextus Pompeius, and finally Antonius against Octavianus.

Naval strength played a key role in each of these affairs. Caesar was nearly defeated in Greece as the Pompeian navy kept Caesarian reinforcements from crossing over from Italy. Sextus Pompeius, the son of Magnus, continued to fight against the Caesarians for years, using tactics similar to those the pirates his father had defeated. (Indeed, it is quite possible some of those pirates Magnus had so graciously resettled ended up joining his son later on.) Finally, the war between Antonius, Cleopatra and Octavianius culminated in the battle of Auctium. (Which wasn't quite as grand an affair as Augustus' propaganda claims it was, but still.) Thus, the fate of the Roman world and the birth of the empire were decided by naval strength.