Question: "Can a Christian today perform an exorcism? What does the Bible say about casting out demons?"
Answer: Exorcism (commanding demons to leave other people) was practiced by various people in the Gospels and the Book of Acts—the disciples as part of Christ's instructions (Matthew 10); others using Christ's name (Mark 9:38); the children of the Pharisees (Luke 11:18,19); Paul (Acts 16); and certain exorcists (Acts 19:11-16).
It appears that the purpose for Jesus' disciples to perform exorcisms was to show Christ's dominion over the demons (Luke 10:17) and to verify that they were acting in His name and by His authority. It also revealed their faith or lack of faith (Matthew 17:14-21). It was obvious that this act of casting out demons was important to the ministry of the disciples. However, it is unclear what part casting out demons actually played in the discipleship process.
Interestingly, there seems to be a shift in the latter part of the New Testament regarding demonic warfare. The teaching portions of the New Testament (Romans through Jude) refer to demonic activity, yet do not discuss the actions of casting them out, nor are believers exhorted to do so. We are told to put on the armor to stand against them (Ephesians 6:10-18). We are told to resist the devil (James 4:7), be careful of him (1 Peter 5:8), and not give him room in our lives (Ephesians 4:27). However, we are not told how to cast him or his demons out of others, or that we should even consider doing so.
Ephesians 6:10-18 outlines clear instructions as to how we are to have victory in our lives in the battle against the forces of evil. The first is placing our faith in Christ (2:8,9), which breaks the rule of "the prince of power of the air" (2:2). We are then to choose, again by God's grace, to put off ungodly habits and to put on godly habits (4:17-24). This does not involve casting out demons, but rather renewing our minds (4:23). After several practical instructions on how to obey God as His children, we are reminded that there is a spiritual battle. It is fought with certain armor that allows us to stand against—not cast out—the trickery of the demonic world (6:10). We stand with truth, righteousness, the gospel, faith, salvation, the Word of God and prayer.
It appears that as the Word of God was completed, the Christians had more weapons with which to battle the spirit world than the early Christians did. The role of casting out demons was replaced, for the most part, with evangelism and discipleship through the Word of God. Since the methods of spiritual warfare in the New Testament do not involve casting out demons, it is difficult to determine instructions on how to do such a thing. If necessary at all, it seems that it is through exposing the individual to the truth of the Word of God and the name of Jesus Christ.