The existence of angels can be hard to fathom or believe because we can't see them. They exist however, and play an important role in the work of God in the world. Some angels spend eternity solely praising God, while others have been given tasks to assist mankind and other physical beings in their existence.
Angels in the Bible
Angels are referred to constantly in the Bible. It is from the Bible that we garner much of the information we have on specific angels and their tasks. The entire book of Tobit is concerned with the Archangel Raphael and his guiding and healing of Tobit's family. The Archangel Gabriel brings messages of miraculous births to Zechariah and Mary (Luke 1). The Archangel Michael is shown in the book of Revelation (Revelation 12) as the leader of the armies of Heaven against the fallen angels.
Angels in the Early Church Fathers
The early Christian writers wrote much about angels. These spiritual beings can be mysterious, since we cannot see them, but the Bible clearly talks of their existence, and so it was important to them to ponder these beings.
...Might I not write you things more full of mystery? But I fear to do so, lest I should inflict injury on you who are but babes (in Christ). Pardon me in this respect, lest, as not being able to receive their weighty import, ye should be strangled by them. For even I, though I am bound (for christ), and am able to understand heavenly things, the angelic orders, and the different sorts of angels and hosts, the distinctions between powers and dominions, and the diversities between the thrones and authorities, the mightiness of Aeons, and the pre-eminence of the Cherubim and Seraphim, the sublimity of the spirit, the Kingdom of the lord, and above all, the incomparable majesty of the Almighty God–though I am acquainted with these things, yet am I not therefore by any means perfect. - Ignatius of Antioch Epistle to the Trallians 5
The thoughts of virtuous men are produced by divine inspiration. The soul is disposed in the way it is, and the will of God is conveyed to human souls, by special divine ministers who assist in such service. For regiments of angels are distributed over nations and cities; and perhaps some even are assigned to particular individuals. - Clement of Alexandria Miscellanies 6, 17, 157, 4
It is proper then to an Angel to minister at the command of God, and often does he go forth to cast out the Amorite, and is sent to guard the people in the way; but these are not his doings, but of God who commanded and sent him. - Athanasius Discourse 3 Against the Arians 25.12
And this conception was a work fulfilled by His Word, and perfected by His Spirit. And so the secondary Splendors came into being, as the Ministers of the Primary Splendor; whether we are to conceive of them as intelligent Spirits, or as Fire of an immaterial and incorruptible kind, or as some other nature approaching this as near as may be. - Cyril of Jerusalem Oration 38 9
When the Son of Man, He says, shall come in His glory, and all the Angels with Him. Behold, O man, before what multitudes thou shalt come to judgment. Every race of mankind will then be present. Reckon, therefore, how many are the Roman nation; reckon how many the barbarian tribes now living, and how many have died within the last hundred years; reckon how many nations have been buried during the last thousand years; reckon all from Adam to this day. Great indeed is the multitude; but yet it is little, for the Angels are many more. They are the ninety and nine sheep, but mankind is the single one. For according to the extent of universal space, must we reckon the number of its inhabitants. The whole earth is but as a point in the midst of the one heaven, and yet contains so great a multitude; what a multitude must the heaven which encircles it contain? And must not the heaven of heavens contain unimaginable numbers? And it is written, Thousand thousands ministered unto Him, and ten thousand times ten thousand stood before Him; not that the multitude is only so great, but because the Prophet could not express more than these. So there will be present at the judgment in that day, God, the Father of all, Jesus Christ being seated with Him, and the Holy Ghost present with Them; and an angel’s trumpet shall summon us all to bring our deeds with us. Ought we not then from this time forth to be sore troubled? - Cyril of Jerusalem Lecture 15 24
And the Son of Man shall confound him, when He shall come in the glory of His Father, with the holy angels.” (Mark viii. 38) The angels come in obedience, He comes in glory: they are His retainers, He sits upon His throne: they stand, He is seated—to borrow terms of the daily dealings of human life, He is the Judge: they are the officers of the court. - Ambrose Exposition of the Christian Faith Book 3, Chapter 13, #106
These Church Fathers speak of the awesome power and place of the angels, yet clearly remind us that they are creatures like us. They have a higher place in creation and a more direct connection with and work from God, but they are clearly inferior to God, being His creatures
Guardian Angels in the Bible
The Catholic Church also teaches that each person is blessed with a guardian angel, who will help and guide that person to protect them from spiritual and physical danger. Jesus speaks of the work of guardian angels. “Take care that you do not despise one of these little ones; for, I tell you, in heaven their angels continually see the face of my Father in heaven.” - Matthew 18:10 In speaking of His care for little children, Jesus mentions that they, and by implication everyone, have an angel to protect them and intercede with God for them. “Are not all angels spirits in the divine service, sent to serve for the sake of those who are to inherit salvation?” - Hebrews 1:14. The author of Hebrews also re-affirms that many angels have been given the task of guarding humans and their affairs. This is a great gift from God, that He sends us spiritual protectors to keep us from harm.
Guardian Angels Early Church Fathers
In the earliest days of the Church, they also understood about the existence and importance of guardian angels.
The thoughts of virtuous men are produced by divine inspiration. The soul is disposed in the way it is, and the will of God is conveyed to human souls, by special divine ministers who assist in such service. For regiments of angels are distributed over nations and cities; and perhaps some even are assigned to particular individuals. - Clement of Alexandria Miscellanies 6, 17, 157, 4
Every believer — although the humblest in the Church — is said to be attended by an angel, who the Savior declares always beholds the face of God the Father. Now, this angel has the purpose of being his guardian. - Origen Homilies on Luke 12
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. - Basil the Great Against Eunomius 3.1
These writers from the first centuries of Christianity spoke very clearly about the existence of and the mission of guardian angels. They knew that these angels had a special role to play and that it is important that we be aware of them, so as to invoke their aid when we need it.