Mrs. O'Dowd is the wife of Major O'Dowd, who leads George and Dobbin's regiment. She is Irish – really, really Irish.
Because the novel is super long and because Thackeray didn't get to go back to the beginning and revise what he wrote to make it match the end (since it had already been serially published), some characters change and evolve in surprising ways. Mrs. O'Dowd starts as comic relief, pure and simple. She's Irish and proud of it. Her speech is written with a really heavy accent, and in her eyes everything that isn't from Ireland is vastly inferior (smaller, uglier, less fancy, less refined, etc.). Hardy-har, those deluded crazies!
But when the army is called to fight, Mrs. O'Dowd is transformed into a model soldier wife. She is brave and resourceful, she keeps it together enough to help her husband go off to war as well as possible, and she tends to the wounds of soldiers coming back from the front.