Gospels of Mathew (6:9–13) and Luke (11:2-4) both mention the prayer as taught by Jesus Christ. Prayer as mentioned in Mathew and Luke differ here and there, but the gist of both is the same. 

According to the Gospel of Mathew, this prayer is taught by Jesus as part of the 'sermon on the mount'. In the Gospel of Luke, this prayer is taught by Jesus to one of the disciples who wanted to know what the right way to pray is.

Πάτερ ἡμῶν ὁ ἐν τοῖς οὐρανοῖς·
ἁγιασθήτω τὸ ὄνομά σου·
ἐλθέτω ἡ βασιλεία σου·
γενηθήτω τὸ θέλημά σου,·
ὡς ἐν οὐρανῷ καὶ ἐπὶ γῆς·
τὸν ἄρτον ἡμῶν τὸν ἐπιούσιον δὸς ἡμῖν σήμερον·
καὶ ἄφες ἡμῖν τὰ ὀφειλήματα ἡμῶν,
ὡς καὶ ἡμεῖς ἀφήκαμεν τοῖς ὀφειλέταις ἡμῶν·
καὶ μὴ εἰσενέγκῃς ἡμᾶς εἰς πειρασμόν,
ἀλλὰ ῥῦσαι ἡμᾶς ἀπὸ τοῦ πονηροῦ.

Our Father who art in heaven,
hallowed be thy name. 
Thy kingdom come.
Thy will be done on earth as it is in heaven.
Give us this day our daily bread, 
and forgive us our trespasses,
as we forgive those who trespass against us,
and lead us not into temptation,
but deliver us from evil.
[For thine is the kingdom, and the power, and the glory, for ever and ever.]
Amen.

Pater noster, qui es in caelis:
sanctificetur Nomen Tuum;
adveniat Regnum Tuum;
fiat voluntas Tua,
sicut in caelo, et in terra.
Panem nostrum cotidianum da nobis hodie;
et dimitte nobis debita nostra,
sicut et nos dimittimus debitoribus nostris;
et ne nos inducas in tentationem;
sed libera nos a Malo

Our Father who art in heaven

Father of us all, who is in heaven

Hallowed be thy name

Your name be taken with reverence, your name is the most sacred and powerful. Let the people not use it lightly, for e.g.: for selfish reasons

Thy kingdom come

This stanza can have two meanings

1. May the whole world be your kingdom (of virtues, of love, of faith)

2. May the spiritual kingdom (of virtues, of love, of faith) come to every heart

Thy will be done on earth as it is in heaven

May this world be (perfect) as you wish for us, as it is (already) in heaven

Give us this day our daily bread

Please provide us with the basic necessities.

[Jesus urges that we pray to our father for even as trivial (but necessary) as daily bread]

And forgive us our trespasses, As we forgive those who trespass against us

Forgive our mistakes (sins, debts), as we forgive those who wrong us. (Empower us to forgive those who sin against us)

And lead us not into temptation

Please don't lead into temptations, don't make us be caught up in the worldly pleasures and forget you.

But deliver us from evil

Protect us from evil. Let us not fall into the clutches of temptations (or Satan)

For thine is the kingdom, and the power, and the glory, forever and ever

O Lord, yours are the kingdoms of heaven and earth, yours is all the power there is, yours is the glory of all victories and achievements (because it was achieved by you not us), it has been this way from ever and will be this way forever.