Symbolism, Imagery, Allegory
Usually an ellipsis ("...") is used in novels to show a hesitation or a pause in dialogue, or to indicate when something is being deliberately left out of the narration. This novel is no exception—the prominent ellipses in Anna Karenina are Tolstoy's version of a bleep or blurred bar. They censor the naughty bits.
The first time comes at the end of Part 2, Chapter 10, when Anna and Vronsky consummate their affair. There's no description of the sex, but the dots stand in where it would have gone, if Tolstoy were a sassier kind of author.
The second time comes in Part 5, Chapter 23, at the end of a sentence that starts, "The doctor told me after my illness…" That's the moment when Anna's telling Dolly that she won't be having any more kids. And again, whatever the doctor actually told Anna is left unclear, but the ellipses make it obvious that something is missing (Anna talking about her means of—gasp!—birth control).
In both cases, all those dots replace a discussion about things that were considered taboo during Tolstoy's time. People didn't want to talk about it, so Tolstoy couldn't explicitly write about it without facing the consequences. But by not saying something and leaving in the dots, the narrator is able to draw attention to the double standards and misconceptions surrounding sex and birth control during this time.