The dosh khaleen are a ruling organization within the Dothraki social hierarchy, specifically the crones who preside over the holy city of Vaes Dothrak. The dosh khaleen are comprised of the widows of slain khals, and serve as seers foretelling futures and interpreting omens for the Dothraki.
Season 1
A Golden Crown
In Vaes Dothrak, Daenerys is summoned by the dosh khaleen to eat a stallion's heart so that the crones can read the omens. At first it seems she won't hold the raw meat down, but to both her and Drogo's relief she is able to compose herself. The crones declare Daenerys and Drogo's son will be 'The Stallion That Mounts The World', a great conqueror whose coming is long foretold in Dothraki history. Daenerys says his name will be Rhaego, for her brother Rhaegar, slain by the usurper Robert Baratheon.
In the books
In the Song of Ice and Fire novels, the dosh khaleen have great honor and status amongst the Dothraki. At their bidding, no blood may be spilled or blade bared within the confines of Vaes Dothrak, though this does not prevent other inventive forms of killing. Daenerys, as the wife of Khal Drogo, is told that when Drogo dies, she will be expected to return to Vaes Dothrak and join the dosh khaleen, on pain of death if she refuses.