The Head of Slytherin is a Professor at Hogwarts School of Witchcraft and Wizardry who has the duty of looking after and punishing the students of their House, Slytherin. The Head of House also serves as a Professor of a subject. Most, if not all, of the heads of Slytherin house attended Hogwarts and were in Slytherin. 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 Salazar Slytherin himself had (determination, resourcefullness, cunning, etc.) and like said above, was probably in Slytherin House when he/she was at Hogwarts.
Severus Snape, Head of Slytherin house during Harry's years at Hogwarts
Salazar Slytherin (c. 1000) Salazar Slytherin established the Slytherin household. He had a very strong belief in the fact that only pure-blood witches and wizards should be trained at Hogwarts School of Witchcraft and Wizardry. Because of this belief, he often had disagreements between the other founders to the school, particularly Godric Gryffindor.
Horace Slughorn (between the 1920s and 1981) Horace was the Potions master, and had a very close relationship with Tom Riddle. He told Tom about Horcruxes, believing the information he gave out would purely be used for educational purposes. When later incidents proved otherwise, Horace retired from his position, presumably because he felt ashamed with himself for the mistake he had made. Horace was well-known for his "Slug Club," a group of talented, well-connected, or famous students he favored.
Severus Snape (between 1981 and 1990 - 1997) Severus was the Potions master at Hogwarts, and served as a Death Eater for awhile. After Voldemort killed Lily Potter, whom Severus had romantic feelings for, Snape served as a double agent, secretly working behind Voldemort's back for Dumbledore. His career ended when Voldemort killed him in order to obtain the Elder Wand. Snape was a difficult teacher, and was disliked by all students except Slytherins. He had a special dislike for Harry Potter because of a grudge against Harry's father. It was also well-known that Snape deeply desired the Defense Against the Dark Arts position, but he never got his chance to teach that class until 1996.
Horace Slughorn (1997 - present) Horace, after being convinced by Dumbledore and Harry Potter, returned to Hogwarts as the Potions master. After Severus killed Dumbledore and became headmaster of Hogwarts, it was decided that Horace was to become the Head of Slytherin House.
