Fourth wall
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The fourth wall is a theatre term referring to the audience. This originates in the idea that there are three walls on a stage: one on the back, one to the left, and one to the right, as well as an imaginary fourth wall in front that contains the players within their play. To "break the fourth wall" means to show awareness of the audience or other things outside of it.
The fourth wall was famously broken in episode 7 of The Daleks' Master Plan, "The Feast of Steven", in which the First Doctor wishes the viewers a "happy Christmas": "Oh and incidentally, a happy Christmas to all of you at home." This is the only case in the series proper in which a character explicitly displays knowledge of being on TV; all other cases of fourth wall breaking involve the characters seeming to talk to or perform for the camera, but nothing that cannot be explained by another character or a mirror being in the position the camera is occupying or a character talking to them self, and nothing which displays knowledge of being a TV character.
At the end of episode 1 and beginning of episode 2 of The Aztecs, Tlotoxl looks directly into the camera as he vows to destroy Barbara.
In episode 2 of The Invasion of Time, the Fourth Doctor breaks the fourth wall by looking at the camera and quipping, "Even the sonic screwdriver won't get me out of this one." At the end of the same serial, he grins mischievously to the camera.
In the final scene of The Caves of Androzani, the Sixth Doctor looks directly at the camera and says "Change my dear; and it seems not a moment too soon".
A running joke throughout most televised stories with Peri Brown during the Sixth Doctor's era would be for Peri to at some point note confusion in her location because of the similar looking passage ways ("All these corridors look the same to me" being the most common, but this could change in different scenarios.) This was a reference to the numerous complains that the BBC production crew would build few sets for corridors and tunnels and could simply change small things about the sets and camera angels to attempt to give the illustration of a larger construct. The added gag was suggested by Nicola Bryant, and has since become a piece of fan lore -- notably used in both The Curse of Fatal Death and The Gunpowder Plot.
At the end of The Trial of a Time Lord's last story, The Ultimate Foe, the Valeyard breaks the fourth wall by looking directly into the camera and laughing.
In Remembrance of the Daleks part 2, as Ace walks out of the B&B, the TV announces, "A new sci-fi series, called Do..." before cutting to the next scene. In the cliffhanger of part 3, after the Dalek ship lands outside Coal Hill School, despite the Doctor's prediction, the Doctor turns to camera and says "I think I might have miscalculated" to the viewer. Unlike other cliffhangers of the classic series, the line is not repeated in the opening of the next episode.
Just before her death in Asylum of the Daleks, Oswin Oswald says, "Run you clever boy. And remember," and turns to the camera.
At the end of The Snowmen, the Eleventh Doctor breaks the fourth wall by saying "Watch me run" to the camera.
In the proms special, Music of the Spheres, the Tenth Doctor breaks the fourth wall by addressing the audience at the Royal Albert Hall.
At the end of the Series 1 finale of K9, The Eclipse of the Korven, K9 breaks the fourth wall by looking directly at the camera and saying, "Affirmative".
In Deep Breath, after the Half-Face Man has fallen from his "escape pod" and been skewered on a spike on the top of the Elizabeth Tower, the Twelfth Doctor looks directly into the camera as the question is posed: "Did the robot self-destruct or is the Doctor a murderer?"
In the Titan back-up comic Wholloween, after the Doctor sees that the TARDIS has been egged and tepeed by angry trick-or-treeters, he notes "This is why I stick to Christmas Specials."
The beginning of Before the Flood features a lengthy segment where the Twelfth Doctor talks directly to the audience and explains the "bootstrap paradox", telling the viewer to Google it. He uses an analogy of how a theoretical time traveler went back in time to meet his hero Ludwig van Beethoven, only to find out he didn't exist, so the time traveler copies down all of Beethoven's music based on his future knowledge, and then publishes them under Beethoven's name. However this means the time traveler was inspired by Beethoven, who was inspired by the time traveler. The Doctor then leaves the viewer with the question "who composed Beethoven's fifth?" before he takes out an electric guitar and plays the Fifth Symphony which transitions to the Doctor Who theme.
