How we cite our quotes:
All happy families are alike, every unhappy family is unhappy in its own way. (1.1.1)
Tolstoy asserts that there is a specific kind of happiness that families can achieve. The infinite variations from that happiness result in a special kind of unhappiness. It's almost as if he thinks there is a single recipe for happy families.
In those days Levin used often to be in the Shcherbatskys' house, and he was in love with the Shcherbatsky household. Strange as it may appear, it was with the household, the family, that Konstantin Levin was in love, especially with the feminine half of the household. Levin did not remember his own mother, and his only sister was older than he was, so that it was in the Shcherbatskys' house that he saw for the first time that inner life of an old, noble, cultivated, and honorable family of which he had been deprived by the death of his father and mother. All the members of that family, especially the feminine half, were pictured by him, as it were, wrapped about with a mysterious poetical veil, and he not only perceived no defects whatever in them, but under the poetical veil that shrouded them he assumed the existence of the loftiest sentiments and every possible perfection. Why it was the three young ladies had one day to speak French, and the next English; why it was that at certain hours they played by turns on the piano, the sounds of which were audible in their brother's room above, where the students used to work; why they were visited by those professors of French literature, of music, of drawing, of dancing; why at certain hours all the three young ladies, with Mademoiselle Linon, drove in the coach to the Tversky boulevard, dressed in their satin cloaks, Dolly in a long one, Natalia in a half-long one, and Kitty in one so short that her shapely legs in tightly-drawn red stockings were visible to all beholders; why it was they had to walk about the Tversky boulevard escorted by a footman with a gold cockade in his hat – all this and much more that was done in their mysterious world he did not understand, but he was sure that everything that was done there was very good, and he was in love precisely with the mystery of the proceedings. (1.6.2)
Levin falls in love with the Shcherbatsky family, in part because he never had a family. As an outsider, Levin seems to think that family is the key to happiness. And it's clear that Levin also sees women as the essential component to a happy family, so it's no surprise that he wants desperately to marry – and specifically to marry Kitty Shcherbatsky.
Marriage had never presented itself to him as a possibility. He not only disliked family life, but a family, and especially a husband was, in accordance with the views general in the bachelor world in which he lived, conceived as something alien, repellant, and, above all, ridiculous. (1.16.5)
Vronsky looks down on normal family life, which maybe explains why he opts for a love affair with a married woman. For a detailed discussion of Vronsky and family, check out his "Character Analysis."