'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.
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, unsurprisingly, 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 Shepard. She mentions that her mother, Matriarch Benezia, never mentioned who her father was, but Liara claims to know her 'second parent' was another asari. This may be speculation on Liara's part as an explanation for Benezia's silence, but it is not unjustified as this is something Benezia would have preferred to keep secret, to avoid the public humiliation. Liara clearly finds discussing her parentage quite painful, but she has wondered about the identity of her 'father' and what happened between her parents.
The asari Justicar Samara, a pureblood herself, theorizes that the stigma against purebloods originates in the increased likelihood of their own daughters being Ardat-Yakshi; asari born with a rare genetic defect as old as the species that causes them to kill the people they mate with. Samara herself had three daughters; all three Ardat-Yakshi.