The Gospels say that Jesus did exorcisms. He Himself fought directly with Satan during the wilderness temptations. When He commissioned His followers, He specifically gave them authority over the spirits ( Mark 6:7,13; Matt 7:11, 10:8; Luke 10:18-20) and specifically over evil forces ( Mark 3:14-15), in His name. In the case of Luke 10, it was not the Twelve disciples who were given the power, but a chosen seventy of Jesus' followers. This amazing authority has been passed on to the Body of believers when they act as a body. It's clear that Jesus saw there to be a battle of some sort against the Devil ( Mark 1:23-28, 3:23; Luke 11:21-23). This was true also of the early church in Acts. That they understood the healing ministry as one made up in part of casting out "unclean spirits", is clear ( Acts 5:16, 8:6-8, 19:12). But many church teachers over the years have taught that these could also be understood as acts of mercy on people who needed it badly, or as acts of psychological release.
The scene in Acts 16:16-18, however, demands to be understood in terms of authority over evil spirits. There, a slave girl has been filled with a spirit of divination (fortune-telling) which made lots of money for her masters. Paul cast this spirit out of her. Satan, of course, fought back through less-supernatural means, the raw greed of her masters. They sent for the authorities, who imprisoned Paul and Silas, which would lead to an earthquake and a witness to the jailer.
Some churches teach that Jesus' exorcisms, as well as His healings, were done to 'show' to everyone Jesus' authority and His true place in the divine scheme of things. There's an element of that (John 2:18, 6:30). But that's not the half of it. There's more! The Synoptic Gospels (Matthew, Mark, and Luke) treat the healings, exorcisms and other acts of power as the start of the Kingdom's arrival. The King was giving orders, and was seeing to it that they were carried out. Fending off Satan was not in the forefront of Jesus' mind; love was. While Jesus did powerful acts to undo Satan's works ( 1 John 3:8), the key motive was compassion for those whom Satan had bound ( Mark 1:41).
Exorcism is an act, not a sign, and a gift only in the sense that it is a power given to the church as an organism and not merely to any one person in it except on behalf of it. It is the power to command the Devil to leave. It is another, more focused side to what some traditions call 'the power of the keys', the power given to the Body of believers, the church, to forgive and to damn. On its other sides it's more normally given to the Body's servants, the ordained ministers. It is something to be set apart from the other matters of the Spirit or the Church, because the Devil is so serious a thing. It calls for intense commitment and even more intense prayer. It's best done as a matter of teamwork, of people acting together as a mini-community because they love that particular person, group of people, or institution. The team's goal is to free that person from their bondage to Satan. It is that love, a love which is Christ's love at work in them, plus the hard work of stripping off the lies, peeling away the layers of self so that this love can get down to where the Devil is lodged within, where the miracle of casting out demons takes place. It is this love which gets whatever goodness there is in that person to join in making the liberation happen.