Sir Walter Elliot was borne three daughters by his wife, Elizabeth, Lady Elliot. The pair were married in 1784, and Elizabeth produced four children: one of these was a boy, who was dead at the time of his birth. The other three were all girls, named Elizabeth, Anne, and Mary.
Austen makes it clear in the text that this was far from an optimal state of affairs for a couple at this point in time. It would have been preferable for Sir Walter to have had a male heir. Without one, he is forced to make alternative arrangements about who will inherit his estate; he is also tasked with looking after and bringing up his daughters and with ensuring that they are well married or otherwise supplied for financially for the rest of their lives.
Sir Walter is not a particularly intelligent or competent gentleman—in fact, Austen describes him as "silly" and rather vain, being more obsessed with his appearance than most women and not a particularly good husband to his wife. As such, after his wife's death, it is quite a difficult task for him to behave as a father to three daughters, with whom he is not really able to connect with or relate to.