To lynch is to murder, or unlawfully kill. When an angry mob kills someone they believe is guilty of a crime, they lynch that person.
Through history, when a group of people murders someone, especially by hanging him by the neck, they are usually said to lynch him. This was unfortunately not uncommon during the Civil Rights struggle of the nineteenth and twentieth centuries in the American South: if a white community believed an African American had done something wrong, they would sometimes lynch him. Lynch is named after Justice Charles Lynch, who condoned mob justice during the American Revolution.