Yes, demons do exist, and, though it is rare, they do sometimes take ‘possession’ of a person’s body. The act by which a demon is driven out of a person is called ‘exorcism’, and remains a practice in the Catholic Church today.
Demons do exist
Demons are fallen angels. They were originally created good by God, but turned away from Him in sin: “The Church teaches that Satan was at first a good angel, made by God: The devil and the other demons were indeed created naturally good by God, but they became evil by their own doing. [...] This “fall” consists in the free choice of these created spirits, who radically and irrevocably rejected God and his reign.” (CCC 391-392)
Humans are subject to the influence of these evil spirits in various ways. Due to Original Sin, all humans are born in "captivity under the power of him who thence had the empire of death, that is to say, the Devil" (Trent, Session 5, Decree Concerning Original Sin, 1). When a person is baptized, he is saved from being subject to Satan and becomes a citizen of the Kingdom of Christ. (O’Donnell)
After being redeemed by Christ, however, we remain subject to temptation: "For we do not wrestle against flesh and blood, but against the rulers, against the authorities, against the cosmic powers over this present darkness, against the spiritual forces of evil in the heavenly places." (Ephesians 6:12) (O’Donnell)
But the influence of the demon, as demonstrated in Scripture and the history of the Church, can go further still. A demon may attack man's body from without, called ‘obsession’, or assume control of it from within, called ‘possession’. (O’Donnell)
Demonic activity in the Bible and in Church History
There is only one possible instance of demonic possession in the Old Testament. In 1 Samuel 16.14 we read: “Now the Spirit of the Lord departed from Saul, and a harmful spirit from the Lord tormented him.” In the New Testament, however, there are innumerable instances of demonic possession. Many exorcisms are recorded, most often peformed by Jesus himself. (O’Donnell)
According to the accounts in the Gospels, the victims were sometimes deprived of sight and speech (Matthew 12.22), sometimes of speech alone (Matthew 9.32; Luke 11.14), sometimes afflicted in ways not clearly specified (Luke 8:2), while, in the greater number of cases, there is no mention of any bodily affliction beyond the possession itself (Matthew 4.24; 8.16; 15.22; Mark 1.32, 34, 39; 3.11; 7.25; Luke 4.41; 6.18; 7.21; 8.2). (O’Donnell)
Further effects are described in other passages. One passage about a young man who is possessed says: “Whenever it seizes him, it throws him down, and he foams and grinds his teeth and becomes rigid. ...And when the spirit saw [Jesus], immediately it convulsed the boy, and he fell on the ground and rolled about, foaming at the mouth.” (Mark 9.18, 22). (O’Donnell)
The possessed are sometimes gifted with superhuman powers: "And when Jesus had stepped out of the boat, immediately there met him out of the tombs a man with an unclean spirit. He lived among the tombs. And no one could bind him anymore, not even with a chain, for he had often been bound with shackles and chains, but he wrenched the chains apart, and he broke the shackles in pieces. No one had the strength to subdue him." (Mark 5.2-4) (O’Donnell)
Some victims described in the Gospels are controlled by several demons (Matthew 12.43, 45; Mark 16.9; Luke 11.24-26); in one case by so many that their name was Legion (Mark 5.9; Luke 8.30). (O’Donnell)
Yet, though the spirits were evil, they could not help testifying to Christ's Divine mission: “And immediately there was in their synagogue a man with an unclean spirit. And he cried out, ‘What have you to do with us, Jesus of Nazareth? Have you come to destroy us? I know who you are—the Holy One of God.’” (Mark 1.23-24; see also Matthew 8.29; 3.12; 5.7; Luke 4.34, 41; 8.28). (O’Donnell)
In the history of the Church since the Ascension of Christ into heaven, evil spirits have continued their efforts to lead people away from God, and the Church has continued to wage battle against them in the name of Christ. This battle has been so salient in the life of the Church that in the 4th century St Athanasius argued that the power of the Sign of the Cross against demons proves the truth of the Christian faith. (On the Incarnation, 31) The lives of the saints from the whole history of the Church frequently include explicit spiritual warfare.
The Church still wages war against evil spirits, which sometimes includes exorcisms: “When the Church asks publicly and authoritatively in the name of Jesus Christ that a person or object be protected against the power of the Evil One and withdrawn from his dominion, it is called exorcism. Jesus performed exorcisms and from him the Church has received the power and office of exorcizing. In a simple form, exorcism is performed at the celebration of Baptism. The solemn exorcism, called ‘a major exorcism,’ can be performed only by a priest and with the permission of the bishop. The priest must proceed with prudence, strictly observing the rules established by the Church. Exorcism is directed at the expulsion of demons or to the liberation from demonic possession through the spiritual authority which Jesus entrusted to his Church. Illness, especially psychological illness, is a very different matter; treating this is the concern of medical science. Therefore, before an exorcism is performed, it is important to ascertain that one is dealing with the presence of the Evil One, and not an illness.” (CCC 1673)
The Church reminds us that Satan, though powerful, is still ultimately subject God: “The power of Satan is, nonetheless, not infinite. He is only a creature, powerful from the fact that he is pure spirit, but still a creature. He cannot prevent the building up of God’s reign. Although Satan may act in the world out of hatred for God and his kingdom in Christ Jesus, and although his action may cause grave injuries—of a spiritual nature and, indirectly, even of a physical nature—to each man and to society, the action is permitted by divine providence which with strength and gentleness guides human and cosmic history. It is a great mystery that providence should permit diabolical activity, but ‘we know that in everything God works for good with those who love him.’ [Romans 8.28]” (CCC 395)
Indeed, “victory over the ‘prince of this world’ was won once for all at the Hour when Jesus freely gave himself up to death to give us his life. This is the judgment of this world, and the prince of this world is ‘cast out.’” (John 14.30; John 12.31, Revelation 12.10)
Catechism of the Catholic Church (CCC)
Council of Trent (Trent)
O'Donnell, M. (1911). Demonical Possession. In The Catholic Encyclopedia. New York: Robert Appleton Company. Retrieved December 3, 2012 from New Advent: http://www.newadvent.org/cathen/12315a.htm
Scripture quotations are taken from the English Standard Version (ESV).