Vastra is a Silurian who was awoken by construction work on the London Underground. She killed some of the workers, in revenge for the deaths of several of her Silurian "sisters", but was eventually persuaded by the Doctor to abandon her vengeance. (This has never been shown on TV but was revealed in dialogue during "A Good Man Goes to War".)
Later (1890s), she lived at 13 Paternoster Row with Jenny Flint (initially her maid but subsequently her wife) &, after the events of "A Good Man...", the Sontaran nurse Strax joined her household as butler-cum-footman. Collectively, the three are often known as the "Paternoster Gang".
Vastra has helped the Metropolitan Police on a number of occasions, becoming known as "the Great Detective". It has been strongly hinted in several episodes that the character Sherlock Holmes (in stories by Sir Arthur Conan Doyle) is based on Vastra, though this is up for debate as Holmes appears as an actual character in several Doctor Who novels, and has also crossed over with Bernice Summerfield in the Big Finish audio dramas range.
She has a longstanding friendship with the Doctor (as established in "A Good Man Goes to War") and is acquainted with River Song as well.