The Ancient of Days is God the Father
Daniel 7:13:
13 “In my vision at night I looked, and there before me was one like a son of man,[a] coming with the clouds of heaven. He approached the Ancient of Days and was led into his presence. 14 He was given authority, glory and sovereign power; all nations and peoples of every language worshiped him. His dominion is an everlasting dominion that will not pass away, and his kingdom is one that will never be destroyed. (ESV)
This closely mirrors Matthew 28:18:
18 And Jesus came and said to them, “All authority in heaven and on earth has been given to me. (ESV)
Further, consider the words of the Angel to Mary in Luke 1:
30 And the angel said to her, “Do not be afraid, Mary, for you have found favor with God. 31 And behold, you will conceive in your womb and bear a son, and you shall call his name Jesus. 32 He will be great and will be called the Son of hthe Most High. And the Lord God will give to him the throne of his father David, 33 and he will reign over the house of Jacob forever, and of his kingdom there will be no end.” (ESV)
So the person coming to the Ancient of Days is Christ. Clearly in Daniel 7 the Ancient of Days is someone Daniel is familiar with, for he doesn't question who the Ancient of Days is, but does question what the meaning of the vision of "the one like the son of man".
As we know, Christ sits at the right hand of the Father, in accord with the above text about Christ being given authority and dominion, as a "right hand man" would be:
Who is to condemn? Christ Jesus is the one who died—more than that, who was raised—who is at the right hand of God, who indeed is interceding for us. (Romans 8:34)
As a last note, notice how the end of Daniel parallels the Book of Revelation and it becomes pretty clear who the Ancient of Days is.
23 “He gave me this explanation: ‘The fourth beast is a fourth kingdom that will appear on earth. It will be different from all the other kingdoms and will devour the whole earth, trampling it down and crushing it. 24 The ten horns are ten kings who will come from this kingdom. After them another king will arise, different from the earlier ones; he will subdue three kings. 25 He will speak against the Most High and oppress his holy people and try to change the set times and the laws. The holy people will be delivered into his hands for a time, times and half a time.26 “‘But the court will sit, and his power will be taken away and completely destroyed forever. 27 Then the sovereignty, power and greatness of all the kingdoms under heaven will be handed over to the holy people of the Most High. His kingdom will be an everlasting kingdom, and all rulers will worship and obey him.’28 “This is the end of the matter. I, Daniel, was deeply troubled by my thoughts, and my face turned pale, but I kept the matter to myself.”
In conclusion, the Ancient of Days is God the Father.