For Mormons, the Holy Spirit (or Holy Ghost) is the third person of the Godhead whose special mission is to guide and teach, testify to God, and comfort and sanctify human souls. In these affirmations, Mormonism is in agreement with mainstream Christianity.
Mormonism departs from mainstream Christianity, however, in teaching that the Holy Spirit is a "one in purpose with the Father and the Son, but is a separate being"  rather than an aspect of God or part of a Trinity.
Unlike God and Jesus, who have physical bodies, the Holy Spirit is pure spirit and has no body. Joseph Smith wrote that God revealed to him that "the Holy Ghost has not a body of flesh and bones, but is a personage of Spirit. Were it not so, the Holy Ghost could not dwell in us." 
- "Who is the Holy Ghost?" Mormon.org.
- Doctrine and Covenants 130:22