Chapter 5: The Holy Ghost, A: The Holy Ghost is the third member of the Godhead

The Holy Ghost is a spirit. He does not have a body, but possesses all the characteristics of a member of the Godhead.

The Father has a body of flesh and bones as tangile as man's; the Son also; but 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.
D&C 130:22

And I said unto him: To know the interpretation thereof--for I spake unto him as a man speaketh; for I beheld that he was in the form of a man; yet nevertheless, I knew that it was the Spirit of the Lord; and he spake unto me as a man speaketh with another.
1 Nephi 11:11

The Holy Ghost is the third member of the Godhead. He is a Spirit, in the form of a man. . . . The Holy Ghost is a personage of Spirit, and has a spirit body only. His mission is to bear witness of the Father and the Son and of all truth. As a Spirit personage the Holy Ghost has size and dimensions. He does not fill the immensity of space, and cannot be everywhere present in person at the same time. He is also call the Holy Spirit, the Spirit of God, the Spirit of the Lord, the Spirit of Truth, and the Comforter.
Joseph Fielding Smith, Doctrines of Salvation, 1:38

The Holy Ghost as a personage of Spirit can no more be omnipresent in person than can the Father or the Son, by by his intelligence, his knowledge, his power, and influence, over and through the laws of nature, he is and can be omnipresent throughout all the works of God.
Joseph F. Smith, Gospel Doctrine, 61

The Holy Ghost knows all things. He must know all things so that he can teach us of them.

Wherefore, watch over him that his faith fail not, and it shall be given by the Comforter, the Holy Ghost, that knoweth all things.
D&C 35:19

And by the power of the Holy Ghost ye may know the truth of all things.
Moroni 10:5

The Holy Ghost testitifies of the Father and Son. As the the third member of the Godhead his chief role seems to be to point towards the Father and Son.

As well as those who should come after, who should believe in the gifts and callings of God by the Holy Ghost, which beareth record of the Father and of the Son.
D&C 20:27

And the Holy Ghost beareth record of the Father and me; and the Father giveth the Holy Ghost unto the children of men, because of me.
3 Nephi 28:11