As the title suggests, a ship captain -- who may also be referred to as a vessel's commanding officer or simply as "captain" – has command of an entire Imperial warship!
Earning command of one's own ship -- regardless of the size of the vessel -- is a position of honour, and has been traditionally considered by many to be the culmination of an Imperial Navy career. Naval history records many incidents of captains turning down promotion to staff positions, and even to more modern or capable ships. The Navy honours these men, as it is a benefit to have captains intimately understand their ship and crew in order to gain the best possible performance from them.
A captain's powers are great -- and his word is law on the bridge of his ship -- having been given much flexibility in style of command, degree of personal interaction with the crew, delegation of duties to his first officer, and preferred operational methods.
Never let it be said, however, that effectively running the day-to-day operations of a warship is not also a highly challenging endeavour... A captain must be in a state of constant activity to ensure the efficiency of onboard operations, and all activities taking place onboard a warship are the captain's ultimate responsibility. His challenges may be different from one day to the next, seeing his ship through battle, and potentially interacting with all ranks of officers onboard.
A captain's primary working relationship is with his first officer... While the first officer typically handles internal fighter coordination, a captain is the official representative of his ship to the Empire, and is also responsible for acting as a communications proxy between his ship and the greater Imperial Navy. (It is via the Captain that the majority of onboard promotions, medal awards, and other formal paperwork will be reviewed.)
If a captain commands a flagship, it is also likely that there will be a flag officer (Systems Force Flag Officer or Sector Group Flag Officer) stationed onboard... A captain is always the primary commanding officer of a ship, however he and the flag officer form a relationship reminiscent of Captain Pellaeon and Grand Admiral Thrawn: the flag officer gives direction to the ship and its subordinate vessels, whereas the captain oversees the details of the ship's day-to-day operations, and may also at times be responsible for formulating the actual orders that will see the demands of the flag officer met.