Fish fingers were a type of human food. They were made from fish cut into rectangles, covered in breadcrumbs, then heated up. Not long after regenerating, the Eleventh Doctor found some frozen fish fingers in Amelia Pond's kitchen, along with some custard. He decided to eat them together as fish custard. (TV: The Eleventh Hour) Fish fingers and custard became an important part of the Doctor's friendship with Amy and Rory.
Later, the Doctor compared House's penchant for eating TARDISes to enjoying processed food like fish fingers. The Doctor's TARDIS, temporarily in human form, expressed puzzlement about what fish fingers were ("Do fish have fingers?"). (TV: The Doctor's Wife)