At the bottom of this post is some information on the O Antiphons, the prayers which are the ancestors of the verses of this hymn.
1. VENI, veni, Emmanuel captivum solve Israel,
qui gemit in exsilio, privatus Dei Filio.
O come, O come, Emmanuel, and ransom captive Israel,The context of this hymn is the exile and captivity of Israel, and the promise of a coming Messiah, the Son of God. This exile need not be confined to their historical captivity among the Assyrians and Babylonians; Israel mourns for lack of Emmanuel up until His coming... and perhaps even now though He has come.
That mourns in lonely exile here until the Son of God appear.
Come, Emmanuel, come: unbind captive Israel,
who, deprived of the Son of God, laments in exile.
R: Gaude! Gaude! Emmanuel nascetur pro te Israel!
Rejoice! Rejoice! Emmanuel shall come to thee, O Israel.I think the common translation fails to capture the sense of the Latin: Emmanuel will not just come to Israel, He will be born for Israel. Yes, He is for all mankind, but His advent is centered upon God's promises to Israel. And so Israel, even in her exile, has cause for rejoicing.
Rejoice! Rejoice! Emmanuel shall be born for you, Israel!
2. VENI, veni, Adonai, qui populo in Sinai
legem dedisti vertice in maiestate gloriae. R.
O come, O come, Thou Lord of Might, Who to Thy tribes on Sinai's heightNote that this hymn is about the coming of Emmanuel. By this verse, the Lord Who gave the Law to Israel at Mt. Sinai is the same Lord Who is Emmanuel, the One Who will be born for Israel. And if He gave the Old Law in maiestate gloriae, how much more glorious will His own coming be?
In ancient times didst give the law in cloud, and majesty, and awe.
Come, Lord, come, Who at the top of Mount Sinai
gave the law to Your people in the majesty of Your glory.
3. VENI, O Iesse virgula, ex hostis tuos ungula,
de spectu tuos tartari educ et antro barathri. R.
O come, thou Rod of Jesse’s stem, from every foe deliver themThe seven verses of the hymn are built around seven titles for the Lord (found in seven prophecies of His coming, received by Isaiah). This title, the "Rod (or Shoot) of Jesse" (Iesse virgula in the hymn, Iesse radix ("Root of Jesse") in the O Antiphons, "virga de radice Iesse" in the Vulgate of Isa. 11:1), was heard in this Sunday's First Reading from Isaiah 11. The verse points to the Lord's power to save His own from their enemies and from the very power of death.
That trust thy mighty power to save, and give them victory o’er the grave.
Come, O shoot of Jesse: lead Your own out from the grasp of their enemies,
and from the sight of hell and the grave of the dead.
4. VENI, Clavis Davidica, regna reclude caelica,
fac iter tutum superum, et claude vias inferum. R.
O come, Thou Key of David, come, and open wide our heavenly home;The prophecy of the key of the house of David (cf. Isa. 22:19ff) is often seen as a precursor to our Lord's words to St. Peter: "I will give you the keys of the kingdom of heaven, and whatever you bind on earth shall be bound in heaven, and whatever you loose on earth shall be loosed in heaven." (Matt. 16:19; cf. "And I will place on his shoulder the key of the house of David; he shall open, and none shall shut; and he shall shut, and none shall open," Isa. 22:22, and "The words of the holy one, the true one, who has the key of David, who opens and no one shall shut, who shuts and no one opens", Rev. 3:7)
Make safe the way that leads on high, and close the path to misery.
Come, Key of David, open up the heavenly kingdom,
make the heavenly road safe, and close up the path of hell.
But the prophecy pertains to Christ as well: Christ is the key Who opens the gate of Heaven to us. In doing so, we pray that He both secure the path to Heaven and bar the road to perdition; for He is the way.
5. VENI, veni O Oriens, solare nos adveniens,
noctis depelle nebulas dirasque mortis tenebras. R.
O come, thou Day-spring from on high, and cheer us by thy drawing nigh;The Lord is called the "Orient": the East, the Daybreak, the rising Sun. Jesus describes His second coming "from the east" like the lightning; His Ascension amid clouds of glory took place to the east of Jerusalem, and the angels assured the disciples that His return would be in the same manner. His coming will be as a light shining on those who have dwelt in darkness. (cf. Isa. 9:2) This is He whom Zechariah, the father of John the Baptist, prophesied, saying, "the day shall dawn upon us from on high to give light to those who sit in darkness and in the shadow of death." (Luke 1:78-79)
Disperse the gloomy clouds of night, and death’s dark shadow put to flight.
Come, O Daybreak, come: comfort us by Your advent;
dispel the dreadful clouds of night and the shadow of death.
6. VENI, veni, Rex Gentium, veni, Redemptor omnium,
ut salvas tuos famulos peccati sibi conscios. R.
O come, Desire of nations, bind in one the hearts of all mankind;I have not found a lyrical version of this verse which translates the Latin; they all appear to draw upon the antiphon, which mentions the "desire of nations", the "cornerstone", and making one of many. This verse, in the Latin, heralds the coming of the King of all nations (consider the Solemnity of Christ, King of the Universe) and the Savior of all men. If I have not translated it incorrectly, the verse draws attention to our sense of sin: we, who are servants of the Lord, are aware of our having sinned against Him. Thus we beg Him come and save us: Hosanna!
Bid Thou our sad divisions cease, and be Thyself our King of Peace.
Come, King of the Nations, come: Redeemer of all, come:
in order to save Your servants, conscious of their own sin.
7. VENI, O Sapientia, quae hic disponis omnia,
veni, viam prudentiae ut doceas et gloriae. R.
O come, Thou Wisdom from on high, Who orderest all things mightily;The last verse acknowledges God as that Wisdom Who orders and ordains the affairs of this world. We wish to have His wisdom, to learn from Him and follow His way, which leads us to His glory.
To us the path of knowledge show, and teach us in her ways to go.
Come, O Wisdom, Who ordains all things here below;
come to show us the way of prudence and glory.
The O Antiphons, which are part of the Divine Office (or Liturgy of the Hours) from December 17th through December 23rd, are the ancestors of this hymn. Here they are in the order they are prayed, one per night:
O Sapientia, quae ex ore Altissimi prodisti, attingens a fine usque ad finem fortiter, suaviterque disponens omnia: veni ad docendum nos viam prudentiae.The first letters of these titles for the Lord, taken in reverse (Emmanuel, Rex gentium, Oriens, etc.) spell "ERO CRAS" in Latin, which means "Tomorrow, I will be (here)", which is very fitting for December 23rd: the next night heralds the birth of Christ.
O Adonai, et dux domus Israel, qui Moysi in igne flammae rubi apparuisti, et ei in Sina legem dedisti: veni ad redimendum nos in brachio extento.
O Radix Jesse, qui stas in signum populorum, super quem continebunt reges os suum, quem gentes deprecabuntur; veni ad liberandum nos, iam noli tardere.
O Clavis David, et sceptrum domus Israel: qui aperis, et nemo claudit; claudis, et nemo aperit: veni et educ vinctum de domo carceris, sedentem in tenebris, et umbra mortis.
O Oriens, splendor lucis aeternae, et sol justitiae: veni, et illumina sedentis in tenebris, et umbra mortis.
O Rex gentium, et desideratus earum, lapisque angularis, qui facis utraque unem: veni, et salva hominem, quem de limo formasti.
O Emmanuel, Rex et legifer noster, expectatio gentium, et Salvator erum: veni ad salvandum nos, Domine Deus noster.
I also recommend reading Dom Prosper Guéranger's commentary on the O Antiphons, a truly amazing resource.