"Pureblood" is a term referring to the child of two asari. It is considered a negatively charged, even offensive word, and emerged due to the unique biology and outlook of the asari.
Background
Since leaving their homeworld and realising they could mate with any species, an attitude has developed among asari that mating with other races gives greater insight into that particular species. Therefore, mating with other races has become the preferred option, since it has an evolving effect on the resulting offspring, and therefore on the asari race itself. "Pureblooded" asari on the other hand, who have two asari parents, are seen as wasteful by some because "nothing has been gained". However, it is considered cruel and insulting to say such a thing to a pureblooded asari's face.
Liara T'Soni talks about herself as a possible pureblood in discussions with Commander Shepard. She mentions that her mother, Matriarch Benezia, never mentioned who her father was, but Liara claims to know that her "second parent" was another asari. This was mostly speculation on Liara's part as an explanation for Benezia's silence, but it was not unjustified, as this was something Benezia would have preferred to keep secret as to avoid the public humiliation. Liara clearly found discussing her parentage quite painful, but she has wondered about the identity of her "father" and what happened between her parents. She does discover that her "father" is indeed another asari.
Ardat-Yakshi are asari born with a rare genetic defect that renders them unable to properly meld with the nervous systems of others, which in its most severe form is lethal to anyone they mate with. Ardat-Yakshi can only be purebloods, which the justicar Samara, a pureblood herself, theorizes is the origin of the stigma against purebloods. Samara herself has three daughters, all of whom are Ardat-Yakshi: Morinth, Rila, and Falere.