In the A Song of Ice and Fire novels, the bulk of the naval forces of the Seven Kingdoms are composed of the royal fleet, the Redwyne fleet, and the Iron Fleet. The royal fleet is anchored at Dragonstone, positioned to defend King's Landing and Blackwater Bay. The large fleet of House Redwyne is used to defend their home ports on the Arbor, a large island off the southwest coast of the Reach. The Iron Fleet, composing the navy of the Iron Islands, isn't usually thought of as at the authority of the Iron Throne. If anything, the royal fleets are often used to keep the Iron Fleet in check when ironborn captains ignore the ban on commerce raiding that the Targaryen kings imposed when they took power. House Lannister also commanded a large fleet anchored at Lannisport, but it was destroyed in the Greyjoy Rebellion.
One of Stannis' few advantages at the opening of the War of the Five Kings is that while he - initially - has the smallest land army out of all factions in the war, as the former Master of Ships, a disproportionately large fraction of the Royal Fleet sides with him. George R.R. Martin confirmed in the Blu-ray commentary for "Blackwater" that while it is not outright stated in dialogue in the TV series, Stannis' position as Master of Ships during his brother Robert's reign is why he has the allegiance of most of the Royal Fleet.