Apart from being Headmaster of Hogwarts, Albus Dumbledore was also the Chief Warlock of the Wizengamot, the Supreme Mugwump of the International Confederation of Wizards, as well as the Founder and Leader of the Order of the Phoenix.
Duties within the Ministry of Magic
The Wizengamot is the High Court of Great Britain and also the Parliament of the Ministry of Magic through which Law and various Decrees are passed. It has fifty members, is located within the Ministry of Magic in London, and it is one of the most important Ministry position in the country. This position is most likely comparable to that of the Chief Justice of the Supreme Court of a Country, one of the most significant appointments only bestowed upon a Wizard who must be extremely worthy of it.
The International Confederation of Wizards is the body which constitutes of Wizarding Delegations (Ministry Seats) from all over the World, and Albus Dumbledore was the Supreme Mugwump. He was responsible for magical co-operation and correspondence between the Ministries of Magic of all the countries in the World. This body is also the key Confederation which enforces and regulates the International Statute of Wizarding Secrecy throughout the World. This position is likely comparable to the muggle-World position of the Secretary General of the United Nations.
Activities with the Order of the Phoenix
The Order of the Phoenix is the only known secret society formed to directly oppose Lord Voldemort and his Death Eaters. During the First Wizarding War, The Order along with Aurors from the Ministry were responsible for secretive, covert operations and assaults towards the Dark Rebellion. (Dumbledore was also responsible for the invention of the Speaking Patronuses). After the rebirth of Voldemort, the Order was alone responsible in intelligence, counter-intelligence and espionage activities within both the Ministry of Magic as well as the Death Eaters. The Order alone planned and initiated Protection of the Prophecy of The Dark Lord/Harry Potter. The Order also was solely responsible for the Protection of Harry Potter throughout the years. At the heart of the Order was Albus Dumbledore, who planned every aspect of the movement and was the leader of all the members.
Activities beyond the Ministry of Magic and the Order of the Phoenix
Infiltration of Voldemort and the Death Eaters
Albus Dumbledore was along with Severus Snape, the only two people who were actively involved in the counter-intelligence operation against Voldemort. Prior to and even after his death, Dumbledore and Snape were the only two people who were fully aware of their operations and none the Order members, nor the Ministry, nor the Death Eaters or Voldemort knew about it. Not even Harry Potter knew about Dumbledore and Snape's exact plans. The two of them spent a huge amount of time planning and conspiring against Voldemort, using Snape as a double-agent to send misleading information to Voldemort, and also finding out about Voldemort's plans. Dumbledore also had planned with Snape about information to pass on to Harry Potter, as well as the protection and assistance of Harry Potter during and beyond his death.
Search for Horcruxes
Albus Dumbledore travelled far and wide in a attempt to research, discover and gather information on Voldemort's past, in order to understand Voldemort's darkest secrets. These trips took him from places like Azkaban, where he retrieved memories from Morfin Gaunt, to Hepzibah Smith's old house-elf Hokey, as well as older Ministry officials like Bob Ogden. Dumbledore also searched for various possible hiding locations for Voldemort's Horcruxes, a result of which he discovered the cave underneath the sea cliff which Voldemort had first visited as a child. Dumbledore also found and destroyed the Horcrux Ring of Gaunt which was hidden in Gaunt's residence.
As Headmaster of Hogwarts, Extended Activities
Albus Dumbledore was amongst one of the only Wizards who ever sought and maintained, on many occasions, relationships with the Centaurs and Mer-people around the Hogwarts Castle. He seems to be the only Wizard besides Hagrid, who has at one time been friends and respected by the Centaurs, and Dumbledore was the only known person to have been in communication with the Mer-people.
It must also be noted that Albus Dumbledore provides a lot of commentary and notes in the book The Tales of Beedle the Bard.