The sign of Emmanuel
'A young woman shall conceive' (Hebrew almah) against the Greek parthenos ('virgin') Matthew cites — the sign to Ahaz in his own crisis and its Gospel application.
Isaiah 7:14
14Therefore the Lord himself will give you a sign. Behold, the virgin will conceive, and bear a son, and shall call his name Immanuel.
Matthew 1:23
23“Behold, the virgin shall be with child, and shall give birth to a son. They shall call his name Immanuel;” which is, being interpreted, “God with us.”
What each tradition says
Rabbinic / Midrashic
Rashi says Immanuel means that our Rock shall be with us. The sign is that she is a young girl who never prophesied, yet divine inspiration shall rest upon her. He cites 8:3, where a prophet's wife is called a prophetess only if she prophesied, and rejects the Hezekiah view because Hezekiah was born nine years before his father's reign.
Rashi (R. Shlomo Yitzhaki, 1040-1105), Rashi on Isaiah (The Judaica Press complete Tanach with Rashi, trans. A. J. Rosenberg) — Comment on Isaiah 7:14 (note on 'Immanuel')
Patristic
Justin Martyr holds the prophecy 'Behold, the virgin shall conceive, and bear a son' was spoken of Christ. If he were not begotten of a virgin, of whom did the Holy Spirit declare it? If he were begotten by sexual intercourse like other first-born sons, why did God call it a sign not common to all first-born sons?
Justin Martyr (c. 100-c. 165), Dialogue with Trypho, ch. 84, in Ante-Nicene Fathers, vol. 1 (trans. Marcus Dods; Buffalo: Christian Literature Publishing Co., 1885) — Chapter LXXXIV ('That prophecy, "Behold, a virgin," etc., suits Christ alone')
Catholic
The Catholic Encyclopedia states that the child of Isaiah 7:14 is neither a son of the Prophet Isaias nor a son of Achaz. It says Isaias 8:1-4 shows the Prophet's son has a different name, and Ezechias did not possess Emmanuel's essential characteristics as Isaias described them. The Emmanuel is the Messias foretold in Isaias's other prophecies.
Catholic Encyclopedia (1907-1912), article 'Emmanuel', The Catholic Encyclopedia, vol. 5 (New York: Robert Appleton Company); New Advent edition — Article 'Emmanuel', section on the identity of the child of Isaiah 7:14
Orthodox
John of Damascus states that the enemy of our salvation kept a watchful eye on virgins. This was according to Isaiah’s prophecy: a virgin shall conceive and bear a Son and shall call His name Emmanuel, interpreted as God with us.
John of Damascus (c. 675-c. 749), An Exact Exposition of the Orthodox Faith, in NPNF ser. 2, vol. 9 (trans. S. D. F. Salmond; T&T Clark, Edinburgh, 1898) — Book IV, ch. 14 (Concerning our Lord's genealogy and concerning the holy Mother of God)
Reformed / Protestant
Calvin says it is unnecessary to contend about the word. Even if it sometimes denotes a young woman and refers to age, Scripture frequently uses it where the subject relates to a virgin. If the Prophet spoke of a young woman conceiving through a man, that would be absurd as a sign or miracle.
John Calvin (1509-1564), Commentary on the Book of the Prophet Isaiah, vol. 1 (trans. William Pringle; Edinburgh: Calvin Translation Society, 1850) — Commentary on Isaiah 7:14
Evangelical
Barnes says the prophecy was originally meant to assure Ahaz that Judah would soon be delivered, that Syria and Israel would lose both their kings, and that this appears to be its literal fulfillment. He adds that it is more difficult to know in what sense it was fulfilled in Christ's birth.
Albert Barnes (1798-1870), Notes on the New Testament, Explanatory and Practical (enlarged ed. by Robert Frew; CCEL carries the Baker reprint) — Note on Matthew 1:22-23 (the quotation of Isaiah 7:14)