The manifestation of discerning of spirits is God or the Lord Jesus Christ revealing to you information about the presence or non-presence of spirits (including both holy spirit or demons), and sometimes including the identity of demons present, whether or not you may cast them out, and providing the power to do it.
Discerning of spirits is a “manifestation” of the gift of holy spirit (1 Cor. 12:10 KJV). Some people are by nature very perceptive and discerning, but that is not the “manifestation of discerning of spirits.” It is not a natural ability. The manifestation of discerning of spirits involves information and power that flows from God or the Lord Jesus to the believer via the gift of holy spirit in that believer. No matter how naturally perceptive a person is, he will not be able to accurately discern the presence of a demon, and will certainly not be able to cast it out. It takes the revelation and power of God to do that.
There are many “spirits” in this world, including angels, evil spirits, and the gift of holy spirit within Christians. Via the manifestation of discerning of spirits, God or the Lord Jesus is able to inform the Christian if a person is saved, unsaved, or under the influence of a demon. Because of the spiritual battle that rages around all of us, the most common aspect of discerning of spirits is dealing with the demonic forces of this world.
Our Adversary, the Devil, walks about as a roaring lion, seeking people to devour (1 Pet. 5:8). God has not left us helpless, but has empowered us to deal with him. The manifestation of discerning of spirits is more than just recognizing them or distinguishing between them, even though “discerning” or “distinguishing” are the most common ways the Greek is translated. The manifestation of discerning of spirits involves more than just knowledge, it involves entering into battle against them and casting them out. The Greek word translated “discerning” (diakrisis) has several meanings. It can mean a “distinguishing” or “differentiation.” Also, it can mean to quarrel.  One of the definitions in Liddel and Scott’s Greek lexicon is “decision by battle, quarrel, dispute.”  In the context of dealing with demons, “discerning” (diakrisis), is much more than just mentally discerning and knowing, it has overtones of quarreling or fighting, and involves our dealing with them via the power of God.
The manifestation of discerning of spirits is interwoven with the other manifestations. For example, a believer manifesting discerning of spirits may be simultaneously aware of the presence of the demon, know what to do about the situation, and begin to command it to come out of the person. Receiving the information about the demon and knowing what to do is similar to, and interwoven with, a message of knowledge and a message of wisdom, while casting out the demon can be in the category of working a miracle (Mark 9:38 and 39), even as a healing can be a miracle (Acts 4:16).
Every Christian will encounter demons, whether he recognizes them or not. What a great blessing and comfort to know that God has equipped each of us to deal with any demon that comes against us. Ephesians 6:12, which says we wrestle with demonic powers, is written to every Christian. Jesus was the first person to regularly and effectively use the power of God to recognize and cast out demons (Matt. 8:16, 9:33, 10:1, etc.), and it is primarily from his example that we learn about discerning of spirits. He taught his disciples about the spiritual battle, and commanded them to cast out demons also (Matt. 10:8; Luke 10:17).
Through the manifestation of discerning of spirits Peter accurately discerned that Satan filled the heart of Ananias (Acts 5:3), and Paul knew that Elymas was a child of the Devil (Acts 13:10). Via discerning of spirits Paul knew that the slave who was mocking them was not just a disgruntled pagan, but someone under the influence of a demon, and he also knew the proper time to cast it out, after having put up with her mocking “many days” (Acts 16:16-18).
The manifestation of discerning of spirits is necessary if Christians are going to deal effectively with the spiritual realities of this fallen world. As with the manifestations of a message of knowledge and a message of wisdom, discerning of spirits is revelation from God, not “guesswork” or “a feeling.” It is wrong and harmful for any of us to say that someone has a demon unless we get actual revelation about it and the guidance to reveal it. Too many Christians mistake their own feelings for “revelation” from God. When it comes to the manifestations of holy spirit, we dare not be like the false prophets who thought their own deep thoughts and feelings were revelation from God (Jer. 14:14). The manifestation of discerning of spirits is one that enables Christians to set captives free. We must use it like Christ did in his life, with love, humility, accuracy, and boldness.
 William Arndt and F. Wilbur Gingrich, A Greek-English Lexicon of the New Testament and Other Early Christian Literature [sometimes called Bauer Lexicon]. (The University of Chicago Press, Chicago, 1979) p. 185.
 Henry Liddell and Robert Scott, A Greek-English Lexicon (Oxford University Press, NY, 1992), p. 399.