Please answer! I have been wondering for ages! I have tried google and all sorts, but can't find the answer.
Part of the answer is found in recognizing the King imagery of a throne and who sits on a King's 'right hand' in ancient monarchies:
Kings place at their right hand those whom they design to honour, or whom they associate with themselves in dominion. No creature can be thus associated in honour and authority with God, and therefore to none of the angels hath he ever said: Sit thou at my right hand. Heb. 1:13. That divine honour and authority are expressed by sitting at the right hand of God, is further evident from those passages which speak of the extent of that dominion and of the nature of that honour to which the exalted Redeemer is entitled. It is an universal dominion. Matt. 28:18; Phil. 2:9; 1 Pet. 3:22; and it is such honour as is due to God alone. John 5:23. (Hodge, Ephesians 1:20)
Of course God the Father is omnipresent and does not literally sit anywhere, so the throne that he sits on is representative of his original authorship and ruling position over all that has been created by his will. The eternal Son also does not literally sit anywhere and along with the Father and the Spirit omnipotently rules over creation, so the question is really about the Trinity and their respective roles in redemption. To get down to the root of the question then, it seems reasonable to assign the imagery of a Father on a throne with his Son on his right hand with the human aspect of the redemption obtained in Christ.
From the gospel we see the Father beginning the process in 'sending' the Son. The eternal Son, entering humanity through the incarnation, was a work to restore creation to its original glory. In Christ's resurrection his human nature was raised up as the Lord of the universe and so those who believe in Christ are actually seated there as well in him:
And God raised us up with Christ and seated us with him in the heavenly realms in Christ Jesus (Ephesians 2:6, NIV)
The only remaining question is what about the Holy Spirit, why is the Spirit not also said to be sitting on a throne? Well, the imagery sort of automatically answers it. The Father sent the Son and through his resurrection lifted the human nature of Christ into a universal lordship over all creation 'by the Spirit'.
I pray that the eyes of your heart may be enlightened in order that you may know the hope to which he has called you, the riches of his glorious inheritance in his holy people, and his incomparably great power for us who believe. That power is the same as the mighty strength he exerted when he raised Christ from the dead and seated him at his right hand in the heavenly realms (Ephesians 1:18-20, NIV)
The Spirit who acts as the immediate presence of the Father through the Son, who performs and perfects the works of God among us is the communicator of the kingdom of God in us, so that even though we still live on the earth, we are mystically united into Christ and experience heaven at God's right hand, inheriting the universe and eternal life. Therefore, the Spirit is filling all in all in the imagery and lifting us up into heaven according to the Father's will. The Father and the Son are 'sitting' and 'sending' the Spirit, therefore, the Spirit can't be sitting in the imagery, as the Spirit is always 'being sent' to ensure God's will is done on earth as it is in heaven.
Jesus and the Father are one God, but they are two persons. Two persons deserve two seats.
To expand slightly, according to mainstream (i.e. Trinitarian) Christianity, Jesus and the Father are separate persons. In fact this passage is considered evidence that Jesus is not simply a different manifestation of God the Father. I recommend consulting a detailed explanation of the Trinity for more information.
Mike gives an excellent answer describing the imagary, by the way.
When asked who he claimed to be, Christ said, "before Moses, I Am." Supposedly he was saying that he is the God of the Old Testament. The God of the Old Testament told Jeremiah that He knew Jeremiah before He formed Jeremiah in his mother's "belly." If we can assume from this that Jeremiah was a spirit in Heaven prior to his mortality, maybe we can assume Jesus was a spirit prior to His mortality and the name of that Spirit was I Am. Maybe, I Am was talking to the Father of Spirits when it was said, "Look, man is as Us, able to tell good from evil." Maybe it was the voice Father of Spirits that was heard saying, "This is my beloved son... " while Jesus was on the Mount of Transfiguration. Maybe the Godhead is comprised of separate individuals and Constantine was a polititian with a bad idea.