Heart

The heart was an organ in many lifeforms. Humans had a single heart while Time Lords had a second heart. Some accounts indicated that a Time Lord would grow his or her second heart after the first regeneration (PROSE: The Man in the Velvet Mask) while others indicated Time Lords had two hearts even in their first incarnation. (AUDIO: Frostfire, PROSE: Time and Relative, TV: The Doctor's Daughter)

If anything made contact with the heart, it could kill a human. Rex Matheson survived a pole in his due to Miracle Day. (TV: The New World) Jack Harkness realised he was immortal when he was shot through the heart and survived on Ellis Island in 1892. (TV: Utopia)

The Great Vampire, lesser vampires and Haemovores could only be killed by a stake through the heart. The more powerful Great Vampires would not be killed unless their heart were completely destroyed, commonly with a bolt of steel. The Time Lord's developed Bowships to destroy the hearts of great vampires. (TV: State of Decay, The Curse of Fenric)

A wave of cubes sent by the Shakri, after being dormant for a year, induced cardiac arrest in a third of Earth's population, as well as the Eleventh Doctor. The Doctor was able to survive with only one heart working, although according to him, the Doctor wouldn't last much longer. Amy Pond restarted his non-functioning heart with a defibrillator. The Doctor was then able to make the cubes reverse the process and save the humans. (TV: The Power of Three)

The Whisper Men could reach into people's hearts and stop them, resulting in death. (TV: The Name of the Doctor)

Alternative definitions
"Heart" could be used figuratively to refer to the physical or ideological centre of a location. Looms were the heart of the Great Houses, (PROSE: Lungbarrow) while the Kandy Man's kitchen was at the heart of Terra Alpha's governance system. (TV: The Happiness Patrol)

Heart could also refer to the change in emotional standing in an individual, that is, having a "change of heart" such as Maldak saving the Governor and Peri Brown from execution on Varos (TV: Vengeance on Varos) or Alan Jackson changing his mind to move away from Sarah Jane Smith. (TV: The Lost Boy)

Behind the scenes
The novel The Man in the Velvet Mask attempted to retroactively explain why the First Doctor appears to only have one heart in the television story The Edge of Destruction. However, the same omission also appears in the Second Doctor television story The Wheel in Space, and the novel Time and Relative very clearly has both the First Doctor and Susan with two hearts. It was not concretely mentioned that the Doctor has a binary vascular system until the Third Doctor story Spearhead from Space.

Later stories would instead show that injury or illness could temporarily stop one of the two hearts. (TV: The Mind of Evil, The Christmas Invasion, The Shakespeare Code, The Power of Three)