"Thing was they bit off a bit more than they can chew with Gran. Little old witch living alone, they probably thought they didn't need to send anyone particularly powerful. Anyway, Dawlish is still in St. Mungo's and Gran's on the run."
-- Neville Longbottom (DH29)
Neville’s grandmother who raised him in the absence of his parents. She is a forceful, strong woman and a powerful witch who wears green robes, a fox-fur scarf, a distinctive hat topped with a stuffed vulture, and carries a large red handbag. Neville loves her, but he’s a bit scared of her as well. She is Frank’s mother and she treats both Neville and his parents very unsentimentally. When Alice gives Neville a bubblegum wrapper, Gran dismisses it and tells Neville to throw it away (Neville doesn’t listen) (OP23).
Neville came into his own after joining the D.A., and Augusta was quoted in the Daily Prophet saying very positive things about him. In the spring of 1998, Death Eaters sent Dawlish to try to take her into custody. It didn’t go so well for Dawlish. As a result, a few weeks later Dawlish was still in the hospital (DH29), while Gran fought in the Battle of Hogwarts (DH31).
Neville says that his grandmother does not think highly of Charms, and would be happier with him pursuing a different subject. According to Professor McGonagall, Augusta Longbottom failed her Charms O.W.L.s. (HBP9)
In, PA7 Neville's boggart assumes the form of Professor Snape, and he imagines Snape in his grandmother's clothes and uses the Riddikulus charm to successfully repel the boggart.
She is friends with Griselda Marchbanks (OP31).
"august" = distinguished, respected, renowned
It seems that Augusta Longbottom and Professor McGonagall must have known each other since McGonagall knew that Longbottom failed her charms O.W.L.s (HBP9). Perhaps they were students together at Hogwarts.