The work of an exhorter among unbelievers is that of sharing the Gospel (Luke 3:18) and calling them to repentance (Acts 2:40).
Among believers, the work of an exhorter is to encourage loyalty to the Lord (Acts 11:23), to prepare the saints for suffering (Acts 14:20), to unite those who are divisive (1 Corinthians 1:10), to call believers to Christ-likeness (1 Thessalonians 2:12), to warn believers of the dangers of sin (Hebrews 3:13), to produce more love within the church (Hebrews 10:25), and to keep the leaders of a church focussed on the work of shepherding the flock (1 Peter 5:1-2).
We read often in the Scriptures of those who exhort others. Dictionaries define exhort along the lines of a strong plea to another, or a firm instruction or passionate argument for some cause. It shares meaning with words like encourage, admonish, an appeal. But it is really the Biblical uses of the word that really add colour to what it means to be an exhorter.
By way of exhortation therefore, I urge you, be an exhorter.