A Roman Catholic friend of mine told me that each person has a guardian angel. He told me it is based on this verse:
Take heed that ye despise not one of these little ones; for I say unto you, That in heaven their angels do always behold the face of my Father which is in heaven. Matthew 18:10 KJV
According to Catholics and Orthodox, is there one guardian angel per person?
It is the belief of both Roman Catholic and Eastern Orthodox that, yes we can.
In Orthodoxy, there is even a prayer to your Angel.
O Holy Angel, keeping guard over my wretched soul and my passionate life, do not forsake me, a sinner, nor depart from me because of my incontinence. Do not give the evil enemy room to overcome me by force of this mortal body. Strengthen my weak and feeble hand, and set me on the way of salvation. Yea, O Holy Angel of God, guardian and protector of my wretched soul and body, forgive me everything by which I have offended you all the days of my life, and even what I have done this past night; protect me during this day, and guard me from every temptation of the enemy, that I may not anger God by any sin. Pray to the Lord for me, that He may confirm me in His fear and prove me a worthy servant of His goodness. Amen. - Orthodox Morning Prayers
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).
Is there one guardian angel for each person?
According to the Catholic Church, Tradition, St. Thomas Aquinas and other theologians, the answer is yes.
St. Thomas states the following in his Summa Theologiae the following:
Whether men are guarded by the angels?
According to the plan of Divine Providence, we find that in all things the movable and variable are moved and regulated by the immovable and invariable; as all corporeal things by immovable spiritual substances, and the inferior bodies by the superior which are invariable in substance. We ourselves also are regulated as regards conclusions, about which we may have various opinions, by the principles which we hold in an invariable manner. It is moreover manifest that as regards things to be done human knowledge and affection can vary and fail from good in many ways; and so it was necessary that angels should be deputed for the guardianship of men, in order to regulate them and move them to good.
Whether each man is guarded by an angel?
Each man has an angel guardian appointed to him. This rests upon the fact that the guardianship of angels belongs to the execution of Divine providence concerning men. But God's providence acts differently as regards men and as regards other corruptible creatures, for they are related differently to incorruptibility. For men are not only incorruptible in the common species, but also in the proper forms of each individual, which are the rational souls, which cannot be said of other incorruptible things. Now it is manifest that the providence of God is chiefly exercised towards what remains for ever; whereas as regards things which pass away, the providence of God acts so as to order their existence to the things which are perpetual. Thus the providence of God is related to each man as it is to every genus or species of things corruptible. But, according to Gregory (Hom. xxxiv in Evang.), the different orders are deputed to the different "genera" of things, for instance, the "Powers" to coerce the demons, the "Virtues" to work miracles in things corporeal; while it is probable that the different species are presided over by different angels of the same order. Hence it is also reasonable to suppose that different angels are appointed to the guardianship of different men.
Whether to guard men belongs only to the lowest order of angels?
As above stated (Article 2), man is guarded in two ways; in one way by particular guardianship, according as to each man an angel is appointed to guard him; and such guardianship belongs to the lowest order of the angels, whose place it is, according to Gregory, to announce the "lesser things"; for it seems to be the least of the angelic offices to procure what concerns the salvation of only one man. The other kind of guardianship is universal, multiplied according to the different orders. For the more universal an agent is, the higher it is. Thus the guardianship of the human race belongs to the order of "Principalities," or perhaps to the "Archangels," whom we call the angel princes. Hence, Michael, whom we call an archangel, is also styled "one of the princes" (Daniel 10:13). Moreover all corporeal creatures are guarded by the "Virtues"; and likewise the demons by the "Powers," and the good spirits by the "Principalities," according to Gregory's opinion (Hom. xxxiv in Ev.).
Whether angels are appointed to the guardianship of all men?
Man while in this state of life, is, as it were, on a road by which he should journey towards heaven. On this road man is threatened by many dangers both from within and from without, according to Psalm 141:4: "In this way wherein I walked, they have hidden a snare for me." And therefore as guardians are appointed for men who have to pass by an unsafe road, so an angel guardian is assigned to each man as long as he is a wayfarer. When, however, he arrives at the end of life he no longer has a guardian angel; but in the kingdom he will have an angel to reign with him, in hell a demon to punish him.
Whether an angel is appointed to guard a man from his birth?
As Origen observes (Tract. v, super Matt.) there are two opinions on this matter. For some have held that the angel guardian is appointed at the time of baptism, others, that he is appointed at the time of birth. The latter opinion Jerome* approves (vide A, 4), and with reason. For those benefits which are conferred by God on man as a Christian, begin with his baptism; such as receiving the Eucharist, and the like. But those which are conferred by God on man as a rational being, are bestowed on him at his birth, for then it is that he receives that nature. Among the latter benefits we must count the guardianship of angels, as we have said above (Articles 1 and 4). Wherefore from the very moment of his birth man has an angel guardian appointed to him.
*St. Jerome is a saint and Doctor of the Church by the Catholic Church, the Eastern Orthodox Church, the Lutheran Church, and the Anglican Communion.
"In the light of this verse, can we infer that there is an guardian angel for each person?"No. That verse has nothing to do with guardian angels. As far as I can remember, the concept of personal guardian angels does not exist in the Bible.