Brigham Young's views of anyone of African descent were extremely racist by modern standards. The primary source for his views is the Journal of Discourses, which contained all of his public lectures, including lectures about race. While quoting all of his statements about race would make this a tremendously lengthy post, I will try to quote those that depict his conception of race and theology.

 

First, he was of the opinion that anyone of African descent was cursed and that the curse was traced back to Cain, who killed his brother, Abel. "Cain slew his brother.... and the Lord put a mark upon him, which is the flat nose and black skin. Trace mankind down to after the flood, and then another curse is pronounced upon the same race – that they should be the 'servant of servants,' and they will be, until that curse is removed". Journal of Discourses, v. 7, pp. 290-291.
  This reference to "servants of servants" is to Genesis 9:18-27, which says, "And the sons of Noah that went forth from the ark were Shem, and Ham, and Japeth: and Ham is the father of Canaan. These are the three sons of Noah: and of them was the whole earth overspread. And Noah began to be an husbandman, and he planted a vineyard: And he drank of the wine, and was drunken; and he was uncovered within his tent. And Ham, the father of Canaan, saw the nakedness of his father, and told his two brethren without. And Shem and Japeth took a garment, and laid it upon both their shoulders, and went backward, and covered the nakedness of their father, and their faces were backward, and they saw not their father's nakedness. And Noah awoke from his wine, and knew what his younger son had done unto him. And he said, Cursed be Canaan; a servant of servants shall he be unto his brethren. And he said, Blessed be the Lord God of Shem; and Canaan shall be his servant. God shall enlarge Japheth, and he shall dwell in the tents of Shem; and Canaan shall be his servant." So Brigham Young viewed those of African descent as being the descendants of Ham (who apparently married a woman of African descent) and being designated "servants of servants". It should be noted that this was not a unique view of Brigham Young of that time, and it was a common belief among many religions of his time and used by southerners as a justification for slavery.

 

As a result of Brigham Young holding this view of a biblical justification for slavery, he was generally not anti-slavery (or pro-abolitionist). "You must not think, from what I say, that I am opposed to slavery. No! The negro is damned, and is to serve his master till God chooses to remove the curse of Ham." Brigham Young, New York Herald, May 4, 1855, as cited in Dialogue: A Journal of Mormon Thought, Spring 1973, p. 56.

 

As a result of these views, Brigham Young instituted a ban on anyone of African descent holding the Priesthood in 1852, which meant that nobody of African descent could hold any leadership position in the LDS Church until that ban was lifted in 1978. They were also prohibited from receiving ordinances that the LDS Church teaches are essential to salvation, because those ordinances require the priesthood. His ban of bestowing the priesthood on anyone of African descent was related to his views of these biblical curses. "That curse will remain upon [anyone of African descent], and they never can hold the Priesthood or share in it until all the other descendants of Adam have received the promises and enjoyed the blessings of the Priesthood and the keys thereof. Until the last ones of the residue of Adam's children are brought up to that favorable position, the children of Cain cannot receive the first ordinances of the Priesthood. They were the first that were cursed, and they will be the last from whom the curse will be removed."

 

As a result of his views on race, he made many strong pronouncements against any intermarriage with someone of African descent by someone who was not of African descent. One of his strongest is as follows: “Shall I tell you the law of God in regard to the African race? If the white man who belongs to the chosen seed mixes his blood with the seed of Cain, the penalty, under the law of God, is death on the spot. This will always be so.” Brigham Young, Journal of Discourses, v. 10, p. 110.

 

You last question about the LDS Church apologizing is an interesting one. The church lifted the ban in 1978, far after most churches in the US had lifted their racists policies (the church never segregated meetings like many US churches did). But there was no apology, just a pronouncement of a new policy. Then, in a recent General conference, one of the church leaders, Dieter Uchdorf, said "And, to be perfectly frank, there have been times when members or leaders in the Church have simply made mistakes. There may have been things said or done that were not in harmony with our values, principles, or Doctrine." That was as close to an acknowledgement of mistakes has come. There has been no specific apology or even a specific denunciation of Brigham Young's and many of his successors' views on race, although they are not taught today. It is problematic for a church that proclaims its leader is a prophet with direct communication with God (and indeed is the mouthpiece of God) to acknowledge mistakes. As a result, most repudiations are extremely vague and unsatisfactory.