The term "Bleak House" refers to two different houses -- the one owned originally by John Jarndyce, to which Ada, Esther, and Richard come to live with him, and to the second Bleak House, built for Esther and her husband at the end of the book. That said, the Bleak Houses in Bleak House are not bleak at all. They are happy houses. The title of the book is more indicative of the social ills and hypocrisy that Dickens addresses in it.
Sir Leicester Dedlock is introduced as the height of British Aristocracy. He has a home in rural Lincolnshire called Chesney Wold. His beautiful wife lives with him, and is the apex of the fashionable world. She hides a terrible secret -- before she met Sir Leicester she bore an illegitimate child with her lover Captain Hawdon. She thinks that child has died.
This child, Esther Summerson, has been raised by her godmother, who she finds out is really her aunt. After this aunt dies Esther is given into the care of a guardian, Mr. John Jarndyce, a kindly man who educates her and brings her to his house to be the companion of his other ward, Ada Clare.
Ada Clare and her distant cousin Richard Carstone live together with Mr. Jarndyce, as wards of the long complicated Chancery suit Jarndyce and Jarndyce. They are thought to be likey beneficiaries of a large inheritance, but the endless nature of the proceedings, and the delays of the Court of Chancery, make this very unlikely. Esther, Ada, and Richard become close friends, and Ada and Richard fall in love.
Lady Dedlock sees a legal paper that her husband's lawyer, Mr. Tulkinghorn, brings to Chesney Wold. She recongnizes the handwriting as her presumed dead lover, Captain Hawdon. Tulkinghorn investigates, and finds out that the the hand is that of Nemo (Latin for "no one"), an impoverished law writer and opium addict who lives above a dillapitated rag-and-bottle shop owned by Mr. Krook. Tulkinghorn discovers Nemo dead. He has no family, and the only person who seems to have anything to say about him is a street orphan called Jo.
Richard Carstone cannot settle on one occupation, and soon becomes engrossed in the Jarndyce and Jarndyce suit, which consumes all his money and his health. He marries Ada Clare in secret. Esther has an admirer in the young Dr. Woodcourt, who loves her even though her face has been scarred by smallpox, which she contracted while caring for Jo, who suffered from the disease. Soon, Esther learns that Lady Dedlock is her mother, and they have one impassioned reunion. They cannot acknowledge one another publically, however, for fear of ruining Sir Leicester's reputation.
Lady Dedlock changes clothing with her maid Hortense, and goes to find Jo. Jo shows her where Hawdon is buried, in a pauper's grave. Tulkinghorn is aware that Lady Dedlock has a secret, and he pursues it ruthlessly. Hortense is dismissed by Lady Dedlock, and, in a vengeful rage, helps Tulkinghorn to find out Lady Dedlock's secret. Tulkinghorn says that he will keep her illegitimate daughter secret out of deference to Sir Leicester, unless Lady Dedlock does something to make him do otherwise. Tulkinhorn had promised to help Hortense find another job, but refuses to do so, so Hortense curses him. Later, Tulkinghorn is shot in his own chambers. John Jarndyce reveals his love for Esther is more than that of a guardian for his ward, and the two become engaged.
Mr. Bucket, a detective, gathers suspects in Tulkinghorn's murder. George Rouncewell, a former soldier and friend of Captain Hawdon, is suspected, as is Lady Dedlock. Sir Leicester is told of his wife's illegitimate daughter, and of her hatred for Tulkinghorn. Shocked, Sir Leicester suffers a stroke. He forgives his wife fully, however. Bucket later finds Hortense to be the murderer, and arrests her.
Because Lady Dedlock doesn't know of the arrest of Hortense and the forgiveness of her husband, in desperation she leaves the house and changes clothes with a poor woman to evade any pursuers. Esther and Detective Bucket attempt to find her, but arrive too late at the gate of the cemetery where Lady Dedlock's lover, Captain Hawdon, is buried. Lady Dedlock has died in the snow.
The case of Jarndyce & Jarndyce is wound up with the surprise discovery of a later will found in Krook's shop. Ada and Richard are indeed the beneficiaries, but there is no money left to inherit. It has all been consumed in legal fees. Richard dies, and Ada gives birth to his son. John Jarndyce realizes that Esther loves Allan Woodcourt more than she loves him, and Woodcourt and Esther are married. They have children, and Ada and Jarndyce are frequent visitors to the new "Bleak House" they create in Yorkshire.