The serpent
Whether the serpent of Eden is a literal animal, the adversary, or a symbolic figure.
Genesis 3:1-14
1Now the serpent was more subtle than any animal of the field which Yahweh God had made. He said to the woman, “Has God really said, ‘You shall not eat of any tree of the garden’?”
2The woman said to the serpent, “We may eat fruit from the trees of the garden,
3but not the fruit of the tree which is in the middle of the garden. God has said, ‘You shall not eat of it. You shall not touch it, lest you die.’ ”
4The serpent said to the woman, “You won’t really die,
5for God knows that in the day you eat it, your eyes will be opened, and you will be like God, knowing good and evil.”
6When the woman saw that the tree was good for food, and that it was a delight to the eyes, and that the tree was to be desired to make one wise, she took some of its fruit, and ate. Then she gave some to her husband with her, and he ate it, too.
7Their eyes were opened, and they both knew that they were naked. They sewed fig leaves together, and made coverings for themselves.
8They heard Yahweh God’s voice walking in the garden in the cool of the day, and the man and his wife hid themselves from the presence of Yahweh God among the trees of the garden.
9Yahweh God called to the man, and said to him, “Where are you?”
10The man said, “I heard your voice in the garden, and I was afraid, because I was naked; so I hid myself.”
11God said, “Who told you that you were naked? Have you eaten from the tree that I commanded you not to eat from?”
12The man said, “The woman whom you gave to be with me, she gave me fruit from the tree, and I ate it.”
13Yahweh God said to the woman, “What have you done?” The woman said, “The serpent deceived me, and I ate.”
14Yahweh God said to the serpent, “Because you have done this, you are cursed above all livestock, and above every animal of the field. You shall go on your belly and you shall eat dust all the days of your life.
What each tradition says
Rabbinic / Midrashic
Jacobs and Blau state that the serpent of Genesis 3 is identified with Satan. As the incarnation of evil, Satan is the arch-enemy of the Messiah; he is Antichrist.
Joseph Jacobs and Ludwig Blau, The Jewish Encyclopedia, s.v. 'Satan' (New York: Funk & Wagnalls, 1901-1906) — article 'Satan', section 'In Talmud and Midrash' — the serpent of Gen. 3 identified with Satan
Patristic
Theophilus of Antioch says Eve was deceived by the serpent in the beginning and became the author of sin. He identifies the wicked demon called Satan as the one who then spoke to her through the serpent and who still works in men possessed by him, invoking Eve.
Theophilus of Antioch (fl. c. 180), To Autolycus, Book II (trans. Marcus Dods), in Ante-Nicene Fathers, vol. 2 (Buffalo: Christian Literature Publishing Co., 1885) — Book II, ch. 28 ('Why Eve Was Formed of Adam's Rib')
Catholic
John Arendzen notes that ecclesiastical usage shows the distinction between demons and the devil in the Fourth Lateran Council’s phrase, “The devil and the other demons.” In this usage, all are demons, and the chief of the demons is called the devil.
John Arendzen, The Catholic Encyclopedia (New York: Robert Appleton Company), vol. 5, s.v. 'Demon' — article 'Demon', opening definition
Orthodox
John of Damascus holds that the one into whose hands God committed the guardianship of the earth was not made wicked in nature. He was good, made for good ends, and received from his Creator no trace whatever of evil in himself.
John of Damascus (c. 676-749), An Exact Exposition of the Orthodox Faith (trans. S. D. F. Salmond), in Nicene and Post-Nicene Fathers, ser. 2, vol. 9 (New York: Christian Literature Publishing Co., 1899) — Book II, ch. 4 ('Concerning the devil and demons')
Reformed / Protestant
Calvin addresses the apparent implication that because Moses says the serpent was crafty beyond other animals, it was induced to deceive man by its own malignity rather than by Satan. Calvin answers that the serpent’s innate subtlety did not prevent Satan from using the animal to effect man’s destruction.
John Calvin (1509-1564), Commentaries on the First Book of Moses Called Genesis, vol. 1 (trans. John King; Edinburgh: Calvin Translation Society, 1847; CCEL text from the Baker reprint) — Commentary on Genesis 3:1
Evangelical
No sourced position found. Checked CCEL: Wesley's Notes (the CCEL set carries no notes on Genesis 3), Matthew Henry's Genesis pages (JS-only shells, no served text), Barnes' Notes (New Testament only); bibliaplus' Adam Clarke commentary (Genesis not hosted); spurgeon.org and Project Gutenberg for a PD sermon or exposition on Gen. iii. 1 — no citable statement on the serpent's identity found.