A promise was a pledge an individual made, to someone else or to themselves, to carry out a certain action or never stray from a given path.
Clara Oswald told the Twelfth Doctor that her request that he not try to take revenge for her murder by the Quantum Shade was not a promise she was asking him to make, but rather an order she was giving him. (TV: Face the Raven)
According to the Eleventh Doctor, the name one chose was "like a promise you make." (TV: The Name of the Doctor) Specifically, in his case, the name of "the Doctor" represented a vow he made to himself: "Never cruel or cowardly. Never give up, never give in." (TV: The Day of the Doctor)
The War Doctor, by engaging in the Last Great Time War and committing terrible acts "in the name of peace and sanity", was considered to have broken this promise, (TV: The Name of the Doctor) and so believed he did not deserve to be called the Doctor. He allowed himself to reclaim his title, just at the end, when the Doctor found another way to end the war, saving Gallifrey. (TV: The Day of the Doctor)
According to a Dalek on Mazam, promises were for humans and were of no use to Daleks. (COMIC: Abslom Daak... Dalek Killer)
According to a Cyber-Leader in the Death Zone, promises made to aliens had no validity. (TV: The Five Doctors)
When presented with Mr Cleaver's findings concerning the planned Blaidd Drwg nuclear power station, "Margaret Blaine" conceded promised that she would stop the project that day, only to reveal herself as Blon Fel-Fotch Passameer-Day Slitheen and decapitate him shortly after. (TV: Boom Town [+]Russell T Davies, Doctor Who series 1 (BBC One, 2005).)