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:
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:
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'.
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 people. Two people 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.