Mormon scripture teaches that God lives on or near Kolob, depending on whether Kolob is a planet or star. Since God is a physical, material being of flesh and bone, he has to live somewhere in the universe and according to Joseph Smith, Kolob is the place. Unfortunately, Kolob doesn't seem to exist.
Kolob as God's Home
In addition to the "home" of God, Kolob is also the "home" of all of God's spirit children — which is to say, all of us. Kolob is where God and his wife or wives create billions of spirit children from the eternal intelligences that exist in the universe. Even animals and plants apparently exist here first as spirits.
According to Mormon teachings, we existed as spirits with God and God's wife or wives for an undetermined amount of time, possibly billions of years, in a state a bliss and happiness. Eventually, though, we made the decision to be born as physical human beings in order to have the chance of progressing and eventually becoming gods ourselves.
Mormon Scripture on Kolob, God's Home
The earliest reference to Kolob is found in the Book of Abraham (3:1-3), which Joseph Smith claimed to have translated from a set of Egyptian Scrolls. According to Smith, the scrolls described a vision of Abraham:
"And I, Abraham, had the Urim and Thummim, which the Lord my God had given unto me, in Ur of the Chaldees;
And I saw the stars, that they were very great, and that one of them was nearest unto the throne of God; and there were many great ones which were near unto it;
And the Lord said unto me: These are the governing ones; and the name of the great one is Kolob, because it is near unto me, for I am the Lord thy God: I have set this one to govern all those which belong to the same order as that upon which thou standest."
Joseph Smith later explained one set of hieroglyphics in the Book of Abraham as describing:
"Kolob, signifying the first creation, nearest to the celestial, or the residence of God. First in government, the last pertaining to the measurement of time. The measurement according to celestial time, which celestial time signifies one day to a cubit. One day in Kolob is equal to a thousand years according to the measurement of this earth, which is called by the Egyptians Jah-oh-eh."
So Abraham took the Urim and Thummim, which were actually two seer stones which Joseph Smith used for treasure hunting, and somehow saw "Kolob," which Mormons believe to be a star or a planet orbiting a star. This is one of many beliefs which Mormons don't typically talk about with non-Mormons, both because it's so at odds with traditional Christianity and because they recognize that it will sound bizarre to almost anyone. You will never see prominent Mormons like Mitt Romney ever mention Kolob at all.
In order to find some sort of real-world evidence that Joseph Smith's revelations and translations were true, many Mormons have tried very hard to find some star or system that could be identified with Kolob in Smith's alleged translation of the Book of Abraham.
One popular focus for a while was to identify some central star or star system in the center of the Milky Way galaxy, presumably based on the assumption that if God is going to have a throne and home planet in our galaxy, then it would be right at the center. No such star or system has ever been identified, though.
Kolob as a Metaphor
Because no actual Kolob has been found and because the idea of God living on a planet somewhere in the galaxy is a bit too bizarre for many people to accept, some Mormons have suggested metaphorical interpretations of the Book of Abraham text. This would allow Mormons to avoid rejecting it entirely without also having to defend the existence of a literal planet where God literally sits on a literal throne.
The most popular metaphorical interpretation seems to be that Kolob is a symbol for Jesus Christ: Jesus is "the great one" — one of many who help govern the universe but who is near to God.
This would be consistent with Mormon teachings about some souls being further along in their development than others, which means that at any given time some are "greater" than others. It would also be consistent with the tradition of referring to Lucifer as a fallen star, something that is traced back to the Old Testament.
Although this metaphorical interpretation of Kolob is plausible, there are two problems with it. First, there is nothing in the text which indicates that it should be read metaphorically. Why read "Kolob" and "stars" metaphorically, but not any of the following text?
The only motivation is incompatibility with modern science and modern sensibilities, which sounds more like rationalization than a rational reading.
Second, the fact remains that Mormons teach that God has a physical, material being made of flesh and bones. If God is physical, then God must have a physical location — he and his wife or wives must spend time somewhere. If not Kolob, then where?