A prophet like Moses
An ongoing prophetic succession, a single awaited prophet (the one of John 1:21), or Jesus in Acts 3 — the passage also states the test of a false prophet.
Deuteronomy 18:15-22
15Yahweh your God will raise up to you a prophet from among you, of your brothers, like me. You shall listen to him.
16This is according to all that you desired of Yahweh your God in Horeb in the day of the assembly, saying, “Let me not hear again Yahweh my God’s voice, neither let me see this great fire any more, that I not die.”
17Yahweh said to me, “They have well said that which they have spoken.
18I will raise them up a prophet from among their brothers, like you. I will put my words in his mouth, and he shall speak to them all that I shall command him.
19It shall happen, that whoever will not listen to my words which he shall speak in my name, I will require it of him.
20But the prophet who speaks a word presumptuously in my name, which I have not commanded him to speak, or who speaks in the name of other gods, that same prophet shall die.”
21You may say in your heart, “How shall we know the word which Yahweh has not spoken?”
22When a prophet speaks in Yahweh’s name, if the thing doesn’t follow, nor happen, that is the thing which Yahweh has not spoken. The prophet has spoken it presumptuously. You shall not be afraid of him.
Acts 3:22-23
22For Moses indeed said to the fathers, ‘The Lord God will raise up a prophet for you from among your brothers, like me. You shall listen to him in all things whatever he says to you.
23It will be that every soul that will not listen to that prophet will be utterly destroyed from among the people.’
What each tradition says
Rabbinic / Midrashic
God's law is perpetual and immutable, and Moses was its promulgator. There can never be another Moses with a different law. Hence, anyone who professes to have received revelations changing the Law is a false prophet.
Solomon Schechter (1847-1915) and S. Mendelsohn, The Jewish Encyclopedia, vol. 10, s.v. 'Prophet, False' (New York: Funk & Wagnalls, 1905) — section 'Criteria' (the religio-moral test)
Patristic
Chrysostom says the promise, “A prophet shall the Lord God raise up for you from among your brethren, like unto me,” would most conciliate them. He observes low and high matters side by side: the one who was to go up into the heavens is like Moses. Yet this was a great thing, for he was not simply like Moses.
John Chrysostom (c. 349-407), Homilies on the Acts of the Apostles, in NPNF ser. 1, vol. 11 (trans. J. Walker and J. Sheppard; Oxford, 1844; ed. Philip Schaff, 1889) — Homily IX (on Acts 3:12), on verses 22-23
Catholic
According to L. W. Geddes, the prophet of Deuteronomy 18:15-22 was also understood as the Messiah, at least at the beginning of our era. He finds this clear from Theudas’s appeal to his gift of prophecy and St. Peter’s use of the passage in Acts 3:22-23.
L. W. Geddes, The Catholic Encyclopedia, vol. 10, s.v. 'Messiah' (New York: Robert Appleton Company, 1911) — Part I, 'The messiah of prophecy' — paragraph on the prophet of Deut 18:15-22
Orthodox
Cyril of Jerusalem says that when the expected Prophet comes, one should recall Jacob’s prophecy to Judah: a prince shall not fail from Judah, nor a ruler from his loins, until He comes for whom it is reserved. He is the expectation not of the Jews but of the Gentiles.
Cyril of Jerusalem (c. 313-386), The Catechetical Lectures, in NPNF ser. 2, vol. 7 (trans. R. W. Church and others; Oxford, 1894) — Lecture XII ('On the words, Incarnate and made man'), section 17
Reformed / Protestant
Calvin concludes that “a Prophet” is used by enallage for a number of prophets. He calls it absurd to restrict it to Joshua or Jeremiah, since Moses is treating the continual manner of the Church’s government, not what God would do within a short time. He adds that applying it strictly to Christ alone is not more correct.
John Calvin (1509-1564), Commentaries on the Four Last Books of Moses, Arranged in the Form of a Harmony, vol. 1 (trans. Charles William Bingham; Calvin Translation Society, 1852) — Commentary on Deuteronomy 18:15-18
Evangelical
Wesley says God will raise up, produce, and send into the world in due time the promised prophet. He identifies this prophet like Moses as Christ, who was truly like Moses in all commendable parts: prophet, king, priest, mediator, excellent in ministry and work, glorious in miracles, and intimate in converse with God.
John Wesley (1703-1791), Explanatory Notes upon the Old Testament (1765); CCEL's 'Wesley's Notes on the Bible' — Note on Deuteronomy 18:15 (with the note on 18:19)