The Quadrireme is the first ranged naval unit available in the game. Although outclassed in melee combat by the Galley, its ranged capabilities allow for "free" attacks, which gives it the overall edge in naval battles. But much more importantly, the Quadrireme is capable of attacking land tiles, and doing significant damage to land units which dare approach the coast!
That makes the Quadrireme an important part of a land invasion when the target civilization mostly has lands along the coast. Despite its limited range, the Quadrireme can make a difference, especially when fighting to take coastal cities.
Barbarians on the coast will usually skip Galleys and move straight to Quadriremes. If you find an outpost on the coast, be ready for a Quadrireme to be lurking nearby. A single Galley will most likely be destroyed in a one-on-one fight, so try to keep them supported with ranged land units or other ships!
Ancient shipbuilders just kept adding banks of oars to the basic design of the galley until they reached the quadrireme, a galley with four banks, each oar rowed by two men. Pliny the Elder reports that no less a scholar than Aristotle ascribed the creation of the quadrireme to the Carthaginians, likely during the 4th Century BC. Their first recorded use was during the siege of Tyre by Alexander the Great, at that time marching south to invade Egypt. After his death, the design proved popular among his successors, squabbling over the empire. During the later Punic Wars, quadriremes were used by both sides. Historical sources (not always reliable) state that these had a displacement of about 60 tons and carried around 75 marines. With their wide, shallow draft, quadriremes were ideal for operations in shallow coastal waters.