This might be better if you're already familiar with the books.
In the opening of the movie, Mikael Blomkvist, a middle-aged publisher for the magazine Millennium, loses a libel case brought by corrupt Swedish industrialist Hans-Erik Wennerstrom. Nevertheless, he is hired by Henrik Vanger, the aged former CEO of a group of companies owned by a wealthy dynasty, in order to solve a cold case - the disappearance, some forty years previously, of Vangers niece when she was sixteen. Lisbeth Salander is a punk computer hacker. Due to problems in her youth she has a curator controlling her money. She needs a new computer, but in order to get it he forces her to have sex with him repeatedly, and with increasing brutality; he threatens to get her into trouble if she tells on him. However, she secretly films the second incident, and later takes revenge by hitting him with a golf stick, undressing him, and tatooing on his chest that he is a rapist. She threatens to reveal the tape to authorities if he does not write favorable reports about her. By accident she meets Blomkvist and helps him with the investigation. Blomkvist is almost killed by a serial killer, but Salander saves him. Finally they discover that the missing woman is alive, in Australia. After they visit her she decides to return, and reunite with Vanger.