Loom

The Loom or Looms is a genetic weaving construction created by Rassilon following the Pythia's curse of sterility on Gallifrey. (NA: Cat's Cradle: Time's Crucible)

They could literally weave a fully grown although not matured Gallifreyan. The Loom also kept tally of all the people it birthed, keeping track of how old each one was and how many regenerations each had gone though.

A Loom was set into each of the Houses of Gallifrey and each House had a specified number of Cousins which could exist in the Family at any given time. (for example, The House of Lungbarrow was allotted forty-five Cousins) When a member of a Family died for the final time, the Loom would weave a new Cousin into the Family. Cases did exist when an additional cousin was woven (such as the Doctor's cousin Owis) but were extremely rare, as such a thing was illegal. (NA: Lungbarrow)

When he was only five years old, the Doctor boasted that he could remember existing in the Loom before he was actually born: "“I can remember waiting to be born…It was like being all strung out. All unravelled inside the Loom. I was spread really thin…I couldn’t think.  Not put thoughts together.  … But I knew where I was and what was happening. I couldn't wait to get out. And then I was born. My lungs nearly burst. The first rush of air was so cold....”"

The truth of this claim is debatable as it's entirely possible the Doctor was simply trying to shock his tutor at the time. (NA: Lungbarrow)

Behind the scenes
The idea that the Looms weave fully grown Gallifreyans appears to conflict with the depiction in the new series of the Master looking like a child at 8 years old. It is possible, although stretching credulity somewhat, that some Looms produce Gallifreyans as children (this would include the Loom of the Master's House, the House of Oakdown) and some as adults (apparently including the Loom of Andred's House, either Redlooms or Deeptree.

The idea of Looms is also difficult to reconcile with both the Doctor's mentions of his parents, and the Master's mention of his father. It is possible there is some sort of adoption system on Gallifrey (meaning the Doctor might not be biologically half-human), or that people could give some of their genetic material to a Loom to become a "parent" (similar to the situation with the Doctor and Jenny).