According to Dictionary.com, "obsessive" is an adjective denoting anything that is, pertains to or resembles obsession. As a noun, the word is used to refer to someone with an obsession or several obsessions.
In terms of psychiatry, the word refers to a person that is continually preoccupied with a particular person, thing or activity or an overriding idea. According to the Online Etymology Dictionary, the psychological sense of "obsession" and related words was first recorded in 1901. The 1670s saw the first use of the word to indicate something that engrosses the mind. The original sense in English came from the French "obsession" as well as directly from the Latin "obsessionem." In this sense, the word indicated a besieging or blockade. Around the beginning of the 17th century, the word was also used to indicate a hostile action taken by an evil spirit.