The Catholic Catechism has something to say on angels:
333 From the Incarnation to the Ascension, the life of the Word incarnate is surrounded by the adoration and service of angels. When God "brings the firstborn into the world, he says: 'Let all God's angels worship him.'"196 Their song of praise at the birth of Christ has not ceased resounding in the Church's praise: "Glory to God in the highest!"197 They protect Jesus in his infancy, serve him in the desert, strengthen him in his agony in the garden, when he could have been saved by them from the hands of his enemies as Israel had been.198 Again, it is the angels who "evangelize" by proclaiming the Good News of Christ's Incarnation and Resurrection.199 They will be present at Christ's return, which they will announce, to serve at his judgement.200
334 In the meantime, the whole life of the Church benefits from the mysterious and powerful help of angels.201
335 In her liturgy, the Church joins with the angels to adore the thrice-holy God. She invokes their assistance (in the funeral liturgy's In Paradisum deducant te angeli... ["May the angels lead you into Paradise. . ."]). Moreover, in the "Cherubic Hymn" of the Byzantine Liturgy, she celebrates the memory of certain angels more particularly (St. Michael, St. Gabriel, St. Raphael, and the guardian angels).
336 From its beginning until death, human life is surrounded by their watchful care and intercession.202 "Beside each believer stands an angel as protector and shepherd leading him to life."203 Already here on earth the Christian life shares by faith in the blessed company of angels and men united in God.
It's clear that it derives much from the pre-Schism writings and liturgies of what is now the Orthodox Church, in the Byzantine Liturgy and the writings of St Basil:
203 St. Basil, Adv. Eunomium III, I: PG 29,656B.
... but a great deal comes from Biblical sources too.
196 Heb 1:6.
197 Lk 2:14.
198 Cf. Mt 1:20; 2:13,19; 4:11; 26:53; Mk 1:13; Lk 22:43; 2 Macc 10:29-30; 11:8.
199 Cf. Lk 2:8-14; Mk 16:5-7.
200 Cf. Acts 1:10-11; Mt 13:41; 24:31; Lk 12:8-9.
201 Cf. Acts 5:18-20; 8:26-29; 10:3-8; 12:6-11; 27:23-25.
202 Cf. Mt 18:10; Lk 16:22; Ps 34:7; 91:10-13; Job 33:23-24; Zech 1:12; Tob 12:12.
There are a number of Biblical verses which speak directly of guardian angels:
Tobit 12 12 [Then the angel said to them] And so, when you and your daughter-in-law Sarah prayed, I brought a reminder of your prayer before the Holy One; and when you buried the dead, I was likewise present with you.
Job 33 23 If there be for him an angel, a mediator, one of the thousand, to declare to man what is right for him ...
Psalm 34 7 The angel of the LORD encamps around those who fear him, and delivers them.
Psalm 91 9 Because you have made the LORD your refuge, the Most High your habitation, 10 no evil shall befall you, no scourge come near your tent. 11 For he will give his angels charge of you to guard you in all your ways. 12 On their hands they will bear you up, lest you dash your foot against a stone. 13 You will tread on the lion and the adder, the young lion and the serpent you will trample under foot.
Matthew 18 10 See that you do not despise one of these little ones; for I tell you that in heaven their angels always behold the face of my Father who is in heaven.
It is from there but a short step which St Basil takes:
All the angels, having but one appellation, have likewise among themselves the same nature, even though some of them are set over nations, while others of them are guardians to each one of the faithful. [Google Books]
St Basil does not say that the faithful have one angel each; but that each of the faithful have a guardian angel (who may also be assigned to watch over others).