An apology was an expression of one's regret for wrongdoing, usually conveyed by saying "sorry".
Rose Tyler apologised to Queen Victoria for her "nakedness", although the Queen forgave her given she was accustomed to raising her daughters. (TV: Tooth and Claw)
K9 Mark 2 gave Starkey a dog whistle, which would call the robot to his side, as an apology for nearly killing him when he mistook the human for a Jixen warrior. (TV: Regeneration)
Vincent van Gogh apologised to the Eleventh Doctor and Amy Pond for the clutter in his house. (TV: Vincent and the Doctor)
Kate Stewart offered the Eleventh Doctor the official apologies of UNIT after the TARDIS was transported by helicopter. (TV: The Day of the Doctor)
The Twelfth Doctor started using apology cue cards due to his sometimes colder traits to get him out of tight situations. (TV: Under the Lake, Face the Raven)
Gaspard de Saux-Tavannes apologised to Catherine de Medici after he mistook her for Simon Duval. (TV: The Massacre)
Daleks did not accept apologies. (TV: The Stolen Earth, AUDIO: Extermination) In the Unbound Universe, Davros apologised to the Fourteenth Black Dalek for abandoning his Daleks, noting that he'd allowed his expectations to blind him to the Daleks' successes. (AUDIO: Masters of War [+]Eddie Robson, Doctor Who Unbound (Big Finish Productions, 2008).)
The Ninth Doctor apologised to Xandra Lenea-Ashford while she was on holiday "for the flowerpot", before giving her a small spade and a flowerpot. (PROSE: Have You Seen This Man?)
Around September 2006, (TV: Boom Town, PROSE: The Time Traveller's Almanac) Mickey Smith apologised on his website for not wanting to wait for Rose anymore, coming to terms with his relationship with her ending. (PROSE: World Saved. Who Cares?)