The Head of Gryffindor House is a Professor at Hogwarts School of Witchcraft and Wizardry with the specific role of looking after the students in their House. It is unknown how a professor would be elected for the post, but it's most likely that the headmaster chooses the professor whom embodies all the qualities Godric Gryffindor himself had (bravery, daring, conscientiousness, etc.) and was in Gryffindor House when he/she was at Hogwarts. Every portrait in Gryffindor Tower depicts a current or former Head of Gryffindor.