Roman fleet commands were essentially the same as army commands--mind you, the same is largely true for Athens, where fleet commanders were usually the elected strategoi (see Phormio, who led both land campaigns and was an extremely successful naval commander). The earliest attested Roman fleet (Livy's belief in a fleet battle around the time of the kings is nonsense--he's misinterpreted the word classis, which by his time usually meant a fleet but in earlier Latin was used to describe the entire citizen body, composed of all social classes. In other words, the army) is during the Punic Wars, but we know next to nothing about the nitty-gritty details of fleet commands until Augustus. Livy speaks of ship crews during the Punic Wars as though they were land units, which is supported by what we know of the Augustan fleet. Under Augustus, individual ships were commanded by the same officers as existed on land, and an entire ship's compliment (including rowers and trained crew, not just marines) were considered soldiers, "milites." So you had centurions and optios and whatnot onboard ship, and the Romans at some point appear to have adopted the Athenian office of trierarch. Fleets were commanded during the Punic Wars usually by promagistrates or serving magistrates. However, magistrates usually had more important things to do (like leading armies--large fleet engagements were much rarer), so already by the Punic Wars Livy mentions the Prefect of the Fleet (praefectus classis), and it appears that by the late Republic fleets were commanded mainly by legates, prefects, and other lower subordinates to a promagistrate. During the Principate these positions mostly became officially codified--the fleet prefect, for example, became one of countless military offices restricted only to equestrians. It's worth noting however, that the Romans didn't actually raise full Roman fleets that often. In the eastern Mediterranean they relied often on their allies to provide contingents and patrol their regions--many of these allies would retain their traditional fleet structure. During the Mithridatic Wars, for example, a large number of Greek allies provided fleets, and we know that the contingent from Cos (from an inscription) retained their traditional Greek command structure.

While naval officers were usually of lower prestige and status, and largely considered a supporting arm (even when led by a magistrate, the main military action was always the one on land), sometimes they were of incredible importance and prestige. Gnaeus Ahenobarbus, the ancestor of the emperor Nero, commanded the Ionian fleet during Philippi and was extraordinarily successful--after the defeat of the Republicans he refused to put down his arms and continuously raided the Ionian coast, becoming in the process almost as much of a problem as Sextus Pompey (his eventual reconciliation with the Triumvirs was a major victory for Antony, although Ahenobarbus would defect to Octavian during Actium). Decimus Brutus commanded Caesar's makeshift fleet brilliantly in Gaul, and took Massalia with the Caesarian fleet during the civil war--for all this he was made promagsterial governor of Gaul. Agrippa trained during his consulship in 37 and commanded it against Sextus Pompey in 36 and again against Antony in 31, where his novel tactics outmaneuvered Antony's fleet--Agrippa was considered, along with Maecenas, Augustus' right-hand man and the greatest military commander outside the imperial family, despite never having led a land campaign.