Jesus is quoting David, the author of Psalm 110, who was inspired by the Holy Spirit to quote the Father speaking to the Son! While that may seem overly complicated, Jesus has a purpose. He is declaring who He is, namely, true God and true Man in one Person.
When the Apostles Creed starts off saying, "I believe in God, the Father" it is important to note that the Second Article begins with the word "And" as we continue to confess who the God is that we believe in. This is a declaration of the Trinitarian faith, that is, trust in the one God who is Father, Son, and Holy Spirit (Gen. 1:26; Is. 48:16; Mt. 28:19).
The Nicene Creed expresses the relationship of the Father and the Son with the phrase "being of one substance with the Father". This means that the eternal Son is not another god, or that He is like God, but not really God; instead, He truly is God, and no one can claim to believe in the Father and not believe in the Son. This is why the Father reveals Himself in the Son (Ps. 110; Jn. 1:18; 14:6,9; Col. 1:15; 2:9).
The Father reveals Himself in His Son concerning His wrath over sin as the Son bears all of our sins and suffers the punishment that should rightly have been ours (Mt. 20:28; Jn. 1:29; Rom. 5:19). At the same time the Father's love for us is made known in the Son's words (Jn. 17:1-5; Lk. 23:34) and actions (Mt. 3:13-17; Heb. 7:26,27; 1 Jn. 2:2).
The Son does all of this by assuming our nature as His own (the "incarnation"); at the same time He retains His own natural godliness The early Church expressed this with some examples: a piece of iron heated in fire remains iron, yet takes on the characteristics of the fire, and, a plain, rough wool garment remains wool even when dyed with a royal purple dye. In both cases the two become one even as the Son is one Person in two natures. The Son of God (true God), is born in time as One who is truly human (with body and soul) by the working of the Holy Spirit in the virgin Mary (Lk. 1:31-35; Gal. 4:4,5).
This is not a science project, but a matter of faith. David speaks of this unique situation by quoting the Father ("the Lord") speaking to the Son ("my Lord") about the glory and power that rightly belongs to Jesus who shares in our nature and yet is true God (Ps. 110). The One descended from David's line ("my Lord; 2 Sam. 7:12; Lk. 1:32; Rev. 22:16; Rom. 9:5) is David's (and our) Redeemer and Savior, the "Seed" of the woman who crushes the serpent (Gen. 3:15) and conquers on our behalf as the true Lord (Rev. 1:8; 12:10,11).