he's the one

For David says concerning him,

“‘I saw the Lord always before me,

for he is at my right hand that I may not be shaken;

therefore my heart was glad, and my tongue rejoiced;

my flesh also will dwell in hope.

For you will not abandon my soul to Hades,

or let your Holy One see corruption.

You have made known to me the paths of life;

you will make me full of gladness with your presence.’

“Brothers, I may say to you with confidence about the patriarch David that he both died and was buried, and his tomb is with us to this day. Being therefore a prophet, and knowing that God had sworn with an oath to him that he would set one of his descendants on his throne, he foresaw and spoke about the resurrection of the Christ, that he was not abandoned to Hades, nor did his flesh see corruption.

Acts 2:25-31

The Jews would not have been happy with the suggestion of an individual resurrection, so Peter shows them that it is prophesied by one of their heroes - David. David died, so all the words of the psalm cannot be directly about him. They must be about God's promised Messiah. Jesus is the only person who has not been held by the grave - he is the Messiah, he is the one you have been waiting for. 

Being therefore exalted at the right hand of God, and having received from the Father the promise of the Holy Spirit, he has poured out this that you yourselves are seeing and hearing. For David did not ascend into the heavens, but he himself says,

“‘The Lord said to my Lord,

“Sit at my right hand,

until I make your enemies your footstool.”’

Let all the house of Israel therefore know for certain that God has made him both Lord and Christ, this Jesus whom you crucified.” (Acts 2:33-36, ESV)

Still talking about Jesus, Peter explains his exalted position and that he has given the gift of the Holy Spirit which has kicked off what is happening there today. Again, wanting to show how this is not a new idea, but the message of the OT, he refers to David's prophecy about "his Lord", the Messiah. The purpose of Peter's speech is that they would know for certain that Jesus is that Lord, that messiah.