The short answer is trireme warfare, although the Romans had far more numerous technological advantages as they ruled during a later period.

Trireme naval combat involved a ram on the front of the ship as the predominent weapon. This required highly maneuverable ships needing to accelerate quickly. The Greeks would align themselves side by side with bows forward then proceed to try and out maneuver their opponents hoping to bring the bow to bear at ramming speed aided by the wind gauge and compliments of rowers. They would aim to ram the stern or broadside in order to disable enemy maneuverablility or smash their weakest point respectively. Once rammed they would row backward to dislodge and hope the enemy ship took on water in order to drown the enemy. These ships featured marines with arrows and javelins in order to target officers but in order to stay fast and maneuverable they wouldn’t hold that many, anywhere from ten to twenty. More marines would be used in battles requiring less maneuverablility like shallow bays and shoals in order to inflict as much damage as possible.

The Romans on the other hand were a land based empire and didn’t feature a prominent navy until the First Punic War as their opponents, Carthage, was a naval super power that preferred to blockade Roman trading over facing their legions on foot. The Romans countered this by redesigning their ships in order to carry larger compliments of soldiers, reinforcing their flanks and inventing the Corvus. This was a large iron bridge with claws at the end designed to fall into Phoenician (Carthage) ships to keep them from pulling back after ramming. Once the Corvus was engaged the two ships would be stuck, allowing Roman troops to board the opposing ship; effectively creating a land battle at sea. By prioritizing a melee at sea the Romans were able to capture many ships intact over sinking them. This was also extremely effective as the Phoenicians used light weight marines like Greece. Rome used this tactic to eventually control the Mediterranean.

Over the centuries both powers had many other ship designs for differing purposes as well as experiments, prototypes and transports. Some of the Roman designs could carry up to 400 men into an engagement called Quinremes. Your question is extremely broad but hopefully that provides a simple base to the function and purpose of Hellenistic warships.