Hey there, I've written like three responses to this since I first saw it come up. So let me start slow and just recommend checking out a few sources first: "Freedom in Exile" the autiobiography of H.H. the Dalai Lama XIV is usually the first source people have interacting with the choosing of a Dalai Lama. It also seems like the primary source material for the Scorcese film "Kundun." Other books about Tibetan history and religion have very intimate details on the choosing of the Tulku tradition, while focusing on the Dalai Lamas since theirs is the most famous Tulku lineage. "The Fourteen Dalai Lamas" by Glenn H. Mullin is a very detailed history from one of the premier Tibetologists. Mullin's work is far from bias, but he's poured over a lot of the historical records in Lhasa and Dharamsala and includes a lot of the records taken during and after the recognition process of the Dalai Lamas. "Secrets of the Vajra World" by Reginald Ray is another good work with a much broader focus, but the last 1/3 really hones in on Tulku tradition and includes the first-person perspective of recognizing Tulkus.

The Dalai Lamas are the most famous of what Tibetan Buddhists refer to as "Tulkus." There's no direct translation, but something along the lines of "recognized reincarnation of a previous teacher" would be the most accurate terminology.

The Dalai Lamas are said to be incarnations of Chenrizi, the Tibetan name for the Indian Avaloketishvara, the Bodhisattva (sometimes referred to as the God) of Compassion. This is far from unique. The Zhabdrungs of Bhutan are also said to be incarnations of Chenrizi. And the Tibetans themselves say there are ten thousand manifestations of Chenrizi at any given time.

That said, we can say the Dalai Lama lineage more or less began with Lama Drom who was a famous Tibetan teacher in the 14th Century. He was instrumental, along with Jey Tsongkhapa, in transforming the Kadampa school of Vajrayana Buddhism into the Geluk school (what Heinrich Harrer refers to as the "Reformed Buddhist Church" versus the Nyingma or "Orthodox Buddhist Church"). Lama Drom died and Tsongkhapa was against tulku lineages, which had begun to pop up ever since the first tulku was recognized: the reincarnation of the Karmapa Lama - the head of the Karma Kagyu school. This is understandable given that the Sakyas (in decline), Kagyus, and Kadampa-Geluks were competing for influence in Tibet at the time.

After Lama Drom died, there was a child born recognized as his reincarnation: Gendun Drup. Gendun Drup was recognized as Lama Drom's reincarnation but no official lineage was established yet. Tulku Lineages are driven entirely by tradition. There is no scripture out there that outlines their recognition or establishment. This creates a lot of problems in the future since the recognition of a tulku can change based on geography and current events.

Gendun Gyatso was recognized as the reincarnation of Gendun Drup. (I'm not sure if there were tests done to confirm this). Sonam Gyatso in his relationship with the Mongol Altan Khan, began a new era of cho-yon relationship between Mongolia and Tibet. ("Cho-yon" is the ideal political system to the Medieval Tibetans. It's commonly referred to as "Priest-Patron" and refers to the Priest, in this case the Dalai Lamas, supporting and teaching the Patron, in this case the Mongol Khans, who then supplies the Priest with funds to support the religious infrastructure and military assistance. The Priest is required to legitimize the Patron's rule in his jurisdiction.)

After recognizing Altan Khan as the reincarnation of Chinggis Khan - thus adding legitimacy to the warlord - Altan Khan recognized Sonam Gyatso as "Dalai Lama." "Lama" - the Tibetan word for teacher. And "Dalai" being the Mongolian word for ocean. It's a direct comparison of the teacher's wisdom to an ocean, a term that can only come from a pair of landlocked cultures.

After this, the process for recognizing Dalai Lamas was somewhat standardized. I say somewhat because it still changed after this. The Fourth all the way through the Seventh Dalai Lamas were recognized through a process that began with either a prophecy from the dying predecessor, a high-ranking-monk's trip to the sacred lake of prophecy Lhamo Lhatso, or both, proceeded with a dramatic search in some of the remotest places in Tibet (in one case the dangerous border region of Monyul, on the border of Bhutan, and in another in a completely foreign country - Mongolia), a dramatic set of tests where the children in question had to recognize their potential predecessor's personal items mixed with the personal items of others, and finally the official recognition and enthronement of the child in Lhasa on the Golden Throne.

When the Qing Dynasty accepted suzerainty over Tibet, they added a step to the process called the "Golden Urn." The Urn - which the Tibetans will always remind everyone was a step not included in the original recognition process, nor was it ever approved of by a Tibetan person - was sent as a gift from the Manchu Emperor and the amban officials insisted on its usage in the recognition of the Dalai Lamas all through the 19th Century.