Oberyn Martell: "So, does this mean I am a 'Master' of something now? Coins, ships?"
Mace Tyrell: "Lord Tywin and I already determined that I shall be the Master of Ships - long before you."
— Mace fails to understand that Oberyn didn't care what specific small council seat he was granted.[src]
Master of Ships is an important position in the government of the Seven Kingdoms. The holder is a member of the monarch's small council, the inner cabinet of the realm. The Master of Ships is charged with commanding the royal fleet and is a member of the small council in King's Landing. Duties of the master of ships include arranging for the building and maintenance of warships, obtaining crews for the warships, and commanding naval operations.
At the start of the TV series, the position is held by Stannis Baratheon, under the monarchy of his brother King Robert. After Robert's death, Stannis rebels against the Iron Throne and the position remains vacant for some time afterwards. After the Purple Wedding, Lord Tywin Lannister awards the position to Mace Tyrell, the Lord of the Reach. This is revealed in a small council meeting when Oberyn Martell asks whether he becomes a master of something now that he is on the council. Mace says that he and Lord Tywin have already determined that he will be the Master of Ships.
When Lord Mace was burned alive alongside his children and many others, the post of the Master of Ships became vacant again.
In the books
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 primarily 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. The Lannisters subsequently rebuilt their local fleet, though it is kept on a purely defensive posture around Lannisport during the War of the Five Kings, due to the nearby threat of the ironborn.
One of Stannis's 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's position as Master of Ships during his brother Robert's reign is why he has the allegiance of most of the Royal Fleet.
At the beginning of the series, Stannis is the Master of Ships. In the third novel the office is given to Mace Tyrell.
In the fourth novel, Cersei renames the office "grand admiral". She removes Mace Tyrell and his bannermen from the council, and names Aurane Waters, a bastard member of House Velaryon, to the seat. Following Cersei's downfall, Aurane flees the capital with the new dromonds she had commissioned.
After Kevan takes over, he appoints Lord Paxter Redwyne as the lord admiral and Master of Ships - a suitable appointment, in view of Lord Paxter's experience as a commander of naval forces.