Blue stockings were a group of intellectual women (and some men) who met together in the 18th century to discuss mostly literary matters, as opposed to meeting together at card parties and routs.

Notable members of the circle were Elizabeth Montague, Hannah Moore Fanny Burney, and Mrs Hester Cahpone .

How did these meeting of intellectual men and women have the term Blue stocking applied to them?

Boswell, in his Life of Dr. Johnson, states that these “bluestocking clubs” were so named because of Benjamin Stillingfleet, who attended in unconventional blue worsted stockings rather than the customary formal black silk stockings.

Much perverse ingenuity was wasted by the writers of the first quarter of the nineteenth century in trying to account for the term “bluestocking.” Abraham Hayward, de Quincey, Mrs. Opie, all sought for an obscure origin in France, in Italy, anywhere, in fact, save where it lay embedded in the writings of the bluestocking circle. The point is still disputed, but critical authorities lean to the Stillingfleet origin, supported by Boswell, and corroborated by Madame d’Arblay. During the annual migration of the great world to Bath, Mrs. Vesey, meeting Benjamin Stillingfleet, invited him to one of her “conversations.” Stillingfleet, the disinherited grandson of the bishop of Worcester, was a botanist and a poet, a philosopher and a failure. He had given up society and was obliged to decline the invitation on the score of not having clothes suitable for an evening assembly. The Irishwoman, a singularly inconsequent person, giving a swift glance at his everyday attire, which included small-clothes and worsted stockings, exclaimed gaily: “Don’t mind dress. Come in your blue stockings.” Stillingfleet obeyed her to the letter; and, when he entered the brilliant assembly where ladies in “night gowns” of brocade and lutestring were scarcely more splendid in plumage than men in garments of satin and paduasoy, the shabby recluse claimed permission to join them by whimsically murmuring: “Don’t mind dress. Come in your blue stockings.”

Stillingfleet was so popular at these conversation parties, that “blew stockings,” as he was called, was in great request. 
“Such was the excellence of his conversation,” wrote Boswell, “that it came to be said, we can do nothing without the blue stockings, and thus, by degrees, the title was established.”

By one of the ironic subtleties of nomenclature, a term originally applied to a man was gradually transferred in deepened tint to the women of these assemblies. It was a name, “fixed in playful stigma,” as one of the circle happily phrased it. For, though bluestockings were estimable women, individually held in high honour, the epithet “blue,” if not a designation of scorn like les femmes savantes, held at least a grain of goodhumoured malice; possibly, because few of them were free from what their “queen,” with frank self-criticism, called, “the female frailty of displaying more learning than is necessary or graceful.”

From: The Cambridge History of English and American Literature in 18 Volumes (1907–21).Volume XI. The Period of the French Revolution.Chapter XV. The Bluestockings.

Of course, as the above quote notes, learned women were not seen as “quite the thing” at this point in history,and eventually the term became almost a term of abuse, and was and still is applied to anyone displaying a pedantic nature.