Since we have gifts that differ according to the grace given to us,
each of us is to exercise them accordingly: if prophecy, according
to the proportion of his faith; if service, in his serving; or he
who teaches, in his teaching; or he who exhorts, in his exhortation...
Romans 12:6-8a
The gift of exhortation is considered one of the speaking gifts, and its Greek root means to come alongside of so as to help. So it is a gift that enables one to give aid to another, to come alongside and give aid by speaking to the need of the moment. It is a gift that enables one to speak truth, God's truth, the truth of Scripture, into the life of a fellow believer. It is the gift to be able to speak the truth, and speak it in love.
Sometimes this will be a word of encouragement and sometimes a word of rebuke, sometimes it will admonish and sometimes it will strengthen, sometimes it will correct and sometimes it will comfort, sometimes it will convict and sometimes it will affirm. It will always aid in helping the fellow believer press on in the faith, to press on in Godliness, to be encouraged to live a more Christlike life, and to see the truth about themselves and the situation they are in.
It is an important and necessary gift to the body of Christ, and is important in discipling and mentoring; and those who exercise this gift should be greatly appreciated.