It is confusing at the start, because you don't know how all the characters tie together (you find out at the end that they do). There's a mystery that they're all connected to, but the point of the mystery is you don't know what it is or whose involved.
Basically, Esther is an orphan, who suddenly is asked to come to be companion to a girl called Ada. Ada and her cousin Richard are both also orphans, and wards in court as a result of a notorious law-suit over a will that has dragged on for years, with most of the money going on lawyers' fees. The young people are all asked to come and live with Mr Jarndyce at his house, Bleak House. On the way, they stay a night with the Jellybe family, where the mother is so obsessed with her charitable projects in Africa that she fails to notice that her husband is a useless idiot and her family is collapsing around her. Esther tries to help the Jellybe children as much as she can, but doens't get very far. The oldest Jellybe girl is in love with a young man with an equally useless, selfish father.
Meanwhile, Lady Dedlock stalks around looking posh and bored. She appears to have some kind of mystery attached to her, which the lawyer is investigating.
Also meanwhile, there are a whole load of poor people who are connected in various ways with the law courts and who are confusing - there's Mrs Whatsit with the birds who's gone mad, there's Mr Krook who is generally unpleasant, there's Mr George with the Shooting Gallery who has a mysterious past, there's Mr Whatsit the scrivener who is hen-pecked, and there's the crossing-sweeper boy, Jo. They're all going to turn out to have clues to the mystery, but you don't know what they mystery is yet.
Hope that helps to some extent! To be honest, thinking back I know the novel well, and still don't know what half the characters are really there for except for to provide pathos / interest and reveal some extra clues.