Whenever you've got a really, really long narrative going, and you decide to fill it with a bazillion different characters, you need to have some kind of shorthand for telling them apart. Sure, names work, but in War and Peace, many of the names are pretty similar. Since so many people back in the day were named after saints, there weren't that many first names out there. Not only that, Tolstoy decided to complicate things by having characters with very similar last names. (Um, Kuragin and Karagin? Really? Why?)
What does Tolstoy do? He goes for the old standby, relying on a few key physical features to help readers remember who's who. So for example, we have Liza Bolkonsky with her slightly curled and slightly hairy upper lip. We have Marya Bolkonsky, who can be either pretty or ugly depending on whether her eyes are luminous or dull. And we have Pierre, who is sometimes fatter and sometimes skinnier, but always aware of his large body. None of these characteristics seem to tell us too much about these people, besides just what they look like, but the highly specific detail makes it easy to remember who's who.
Speech and Dialogue
Characters who don't have some easily identifiable physical characteristic get a verbal tic to help us remember them by. We have Denisov, with his strange garbling of the letter "r" – depending on the translation you get, it'll either be written as a speech impediment where he can't pronounce the sound, or as some kind of crazy over-revving of his tongue every time he says a word with an "r" in it. In a slightly different vein, we have Julie Karagin, who is so Frenchified that she almost can't speak or write Russian. Every time we hear her use what should be her native tongue, she puts out totally garbled and hilariously non-grammatical nonsense. Finally Tolstoy puts in a lot of effort to mark the speech of the uneducated peasants we come across as really, really different. They speak in confused, chaotic, unstructured sentences, and their language is usually full of colorful local phrases and sayings.
Thoughts and Opinions
The main way we learn about characters, as in most novels, is by hearing their internal monologue. The long sequences where Pierre is having one crisis of faith after another let us know that he's a questioner and a restless ball of anxiety. Contrast his inner voice with Andrei's thoughts. Andrei is never shown to have any doubts, he never second-guesses his decisions or his opinions. We see from the inside out just how much he is like his inflexible, idealistic, and totally obstinate father. Being able to follow the thoughts of the characters – thoughts that don't seem to be edited or judged by the narrator – gives us the ability to see that the dramatic transformations the characters undergo make psychological sense and have an internal logic.