Question number 8 of the most commonly asked Questions about Genesis is: "Did I understand you to say the Orthodox don't believe in inherited (original) sin? If so, how you they explain David saying he was conceived in sin, and sinful from birth?"
In response, I defer to one who is better qualified to answer the first part of this question. The Very Reverend Antony Hughes, M.Div., is the rector of St. Mary’s Orthodox Church in Cambridge, Massachusetts. He has served as the Orthodox Chaplain at Harvard University. Father Hughes has written an excellent explanation of the difference between original sin and ancestral sin. Here is an excerpt:
"As pervasive as the term original sin has become, it may come as a surprise to some that it was unknown in both the Eastern and Western Church until Augustine (c. 354-430). The concept may have arisen in the writings of Tertullian, but the expression seems to have appeared first in Augustine’s works. Prior to this the theologians of the early church used different terminology indicating a contrasting way of thinking about the fall, its effects and God’s response to it. The phrase the Greek Fathers used to describe the tragedy in the Garden was ancestral sin.
Ancestral sin has a specific meaning. The Greek word for sin in this case, amartema, refers to an individual act indicating that the Eastern Fathers assigned full responsibility for the sin in the Garden to Adam and Eve alone. The word amartia, the more familiar term for sin which literally means “missing the mark”, is used to refer to the condition common to all humanity (Romanides, 2002). The Eastern Church, unlike its Western counterpart, never speaks of guilt being passed from Adam and Eve to their progeny, as did Augustine. Instead, it is posited that each person bears the guilt of his or her own sin. The question becomes, “What then is the inheritance of humanity from Adam and Eve if it is not guilt?” The Orthodox Fathers answer as one: death. (I Corinthians 15:21) “Man is born with the parasitic power of death within him,” writes Fr. Romanides (2002, p. 161). Our nature, teaches Cyril of Alexandria, became “diseased… through the sin of one” (Migne, 1857-1866a). It is not guilt that is passed on, for the Orthodox fathers; it is a condition, a disease.
In Orthodox thought Adam and Eve were created with a vocation: to become one with God gradually increasing in their capacity to share in His divine life — deification (Romanides, 2002, p. 76-77). “They needed to mature, to grow to awareness by willing detachment and faith, a loving trust in a personal God” (Clement, 1993, p. 84). Theophilus of Antioch (2nd Century) posits that Adam and Eve were created neither immortal nor mortal. They were created with the potential to become either through obedience or disobedience (Romanides, 2002)."
Read more here.
In answer to the second part of the question: why David said he was conceived in sin and sinful from birth - he is speaking of ancestral sin. This is significant since it was likely during King David's reign that the geneological information in Genesis and Exodus was complied.
Why so much interest in his ancestors? Because David believed the prophecy concerning the coming of the Son of God from his line of ancestors (Gen. 3:15) - He who would crush the head of the cosmic serpent. So the Orthodox proclaim that the Son of God, born of the line of David, reversed the curse that fell upon His and our ancestors.
This is the heart of our catholic Faith: Jesus Christ, the Son of God, has come into the world to save sinners such as me. The Eternal Son became flesh and in His death He destroyed death and the works of the devil.
"For God so loved the world that He sent His only Begotten Son, that whoever believes in Him should not perish but have everlasting life. Fof God did not send His Son into the world to condemn the world, but that the world thorugh Him might be saved."
Three bear witness on earth to the appearing of the Son of God: the Spirit, the water and the blood (I John 5:8). In real time these are Anna the Prophetess (Spirit), John the Forerunner (water), and Simeon the Priest (blood)
“He who believes in the Son of God has the witness in himself; he who does not believe God has made Him a liar because he has not believed the testimony God has given of His Son. This is the testimony: that God has given us eternal life, and this life is in His Son. He who has the Son of God has life; he who does not have the Son of God does not have life.” (I John 5, 10, 11)
Justin Martyr: “There is not a single race…among whom prayers and thanksgivings are not offered in the name of Jesus…”
Irenaeus: “Such is the common faith and tradition…In whom have all the nations believed, but in the Christ…”
The testimony of Scripture: “These things I have written to you who believe in the name of the Son of God, that you may know that you have eternal life and that you may continue to believe in the name of the Son of God.” (I John 5:13)
Jesus came to "trample down death by death" and to overturn the curse that fell upon all the creation when Eve, the crown of creation, subjected herself to the will of the serpent, the lowest of creatures in the hierarchy of creation.
Related reading: Hierarchy in Creation: The Biblical View; The Biblical Meaning of Eve: St. John Chrysostom on Eve's Sin