Who was the author? The Holy Spirit inspired the book of 3 John through St. John, apostle and evangelist, who again refers to himself as "the elder".
What is the book? The book of 3 John in at least some sense is a "catholic" or "universal" apostolic epistle. Despite the fact that it was directly addressed to a Christian named Gaius (probably not the same as the other New Testament figures with this name), the letter is indirectly addressed to a church's leader named Diotrephes and his whole congregation.
Where was it written? Like the other two epistles of St. John and the Gospel account that bears his name, 3 John was likely written from Ephesus.
When was it written? The somewhat wide range of dates for 3 John is the same as for John's other two epistles, A.D. 90 to 100 or 85 to 95, with some theorizing that the letter referred to in 3 John 9 is in fact 2 John.
Why? John had been warning the believers not to welcome false teachers, but, when Gaius welcomed a true teacher sent by John, Diotrephes tried to put Gaius out of their congregation. John writes to support Gaius and warn Diotrephes.
How? John's support for Gaius and warning to Diotrephes is quite clear. If indeed 2 John and 3 John were written and delivered at the same time, John quite shrewdly exercises his apostolic authority, even promising to visit in person.
For further reading on the book of 3 John:
- Kretzmann, Paul E. Popular Commentary of the Bible: The New Testament, Volume II, The Epistles of the Apostle Paul, The Epistle to the Hebrews and the Catholic Epistles, the Book of Revelation. St. Louis: Concordia Publishing House, 1922. (There are 3 pages of comments on 3 John in this commentary.)
- Lenski, R. C. H. The Interpretation of the Epistles of St. Peter, St. John and St. Jude. Minneapolis, MN: Augsburg Publishing House, 1961. (There are 17 pages specifically on 3 John in this commentary.)