The Family of Blood were a family of hunters who hunted the Tenth Doctor in order to take his near-immortality. They tended to refer to one another by their relationship to each other within the family. For example, the patriarch of the Family was called "Father of Mine" by his children, and "Husband of Mine" by his wife.
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Biology
The Family of Blood were green, gaseous predators that could inhabit and control human bodies. Before taking human form, the members of the Family were kept in small, spherical receptacles. (TV: Human Nature) They had very short lifespans, and would naturally perish in just three months, "like mayflies". (TV: The Family of Blood)
While possessing humans, the Family could telepathically communicate with each other from a distance. They could also access the minds and memories of their hosts, though these were mostly obliterated during the possession process, making it difficult for the Family to successfully pose as their victims without arousing suspicion.
They also had an incredible sense of smell, tracking the Doctor wherever the TARDIS took him by his scent alone. (TV: Human Nature)
Technology
The Family utilised a spacecraft that held their 'essence' within it prior to them assuming human form. The interior contained green lights, pillars and paddle-like switches and seaweed-like plants dangling from the ceiling. The spacecraft had a considerable arsenal of technology — the most notable of which was a vortex manipulator (stolen from a Time Agent) that allowed the craft to travel through time. It could also project force fields, render itself invisible and had bomb-shooting cannons.
On hand, the Family of Blood used energy guns that fired green energy that disintegrated humanoid tissue — the weapons apparently consisted of a living creature within a shell that shot the beam when pierced by the trigger. They were somewhat difficult for humanoids to hold.
The Family also had scarecrow-like soldiers, created by Son of Mine, that were placed in various fields. They were composed of straw, their molecules animated via a scientific process. Even after being shot down, they could still reanimate by the command of the Family. (TV: Human Nature / The Family of Blood)
History
The Family encountered the Tenth Doctor and Martha Jones, and chased them throughout time and space with a stolen vortex manipulator, hoping to steal from the Doctor some means of ensuring that they would live a longer lifespan. (TV: Human Nature) According to Son of Mine, on the Family's travels through time and space in search of the Doctor, they saw many things, including the First World War. (TV: The Family of Blood)
The Doctor was forced to turn himself into a human using the Chameleon Arch in his TARDIS in order to escape them. The Doctor changed himself into John Smith, a teacher at a boy's school in England, 1913. The Family eventually tracked him there and landed nearby. They then used animated scarecrows to round up humans for the Family to possess. (TV: Human Nature)
The Family's quests for immortality ended when the Doctor was returned to his Time Lord state with the help of Martha, school nurse Joan Redfern, and student Tim Latimer. He fooled the Family by pretending to still be John Smith and using an olfactory misdirection to trick them into thinking that he was still human. Moments later, an act devised to look like a clumsy accident by "John Smith" resulted in the Family's ship overheating and exploding. The Doctor subsequently punished the Family, each receiving different punishments;
- Father of Mine was wrapped in supposedly unbreakable chains forged from a dwarf star alloy, and left in an underground chamber.
- Mother of Mine was thrown into the event horizon of a collapsing galaxy, imprisoning her there for all eternity.
- Daughter of Mine was trapped within every mirror in existence, where the Doctor would visit her once a year.
- Son of Mine was suspended in time, and dressed as a scarecrow to watch over the fields of England as their protector.
We wanted to live forever. So the Doctor made sure that we did.
In a sense, the Family's quest for immortality was won, although not in the manner they wished. They also discovered from this why the Doctor decided to flee rather than fight them — he was showing them mercy by allowing them the chance to live out their lives then die, rather than killing them or dooming them with immortality. (TV: The Family of Blood)
However, every member of the Family apart from Mother of Mine managed to later escape their imprisonment with the help of Daughter of Mine, who managed to escape first from her entrapment in the mirror. It was implied that she first broke out Son of Mine from his temporal suspension then managed to break the dwarf star alloy chains which were holding Father of Mine. However, the Doctor later managed to recpature the Family, and placed Father and Son of Mine into the event horizon of the collapsing galaxy to be trapped alongside Mother of Mine, and trapped Daughter of Mine inside a glass bottle within the TARDIS. (PROSE: Blood Will Out)
All these events were one iteration of a "story" which played out many times, each time differently. In another iteration of the story, the Seventh Doctor was pursued by the Aubertides, who filled the same role as the Family of Blood. (WC: Shadow of a Doubt)
At least some members of the Family of Blood were entombed on Occasus and being exploited by the Incorporation. (AUDIO: Planet of the End)
Other references
The Twelfth Doctor expressed regret for bringing the Family of Blood to England. (AUDIO: Dead Media)
Behind the scenes
The Visual Dictionary
The non-narrative source Doctor Who: The Visual Dictionary gave further information about the Family of Blood. Their telepathic communication means that when in very high trauma, others can feel this pain. When taking over human bodies, the Family's gaseous form enters them through the eyes.