The book was written by “The Elder” and addressed to the Elect Lady and her Children. Who is “the Elder”? The traditional answer understands the elder as the same person who wrote 1 John, the Apostle John, son of Zebedee. This traditional answer is often challenged based on the date of the letter. If it was written in the early 90s then John the son of Zebedee would be very old.
Eusebius reports a tradition from Papias that someone named John the Elder was active in Ephesus at the end of the first century and many scholars think the tradition John the Apostle settled in Ephesus has been confused with the activity of this John the Elder. By way of analogy, the Philip who was martyred in Hierapolis is called the apostle, but he also had four daughters who were prophets. This is a confusion of Philip the Apostle and Philip the Deacon.
The word translated elder (πρεσβύτερος) refers to a man in his early to mid-50s, as opposed to an old or elderly man (γέρων). But the word was used for a position of authority in the early church. The leaders of a Jewish synagogue and the Sanhedrin were called elders (BDAG), as were the leaders of towns (BDAG, citing LXX Ruth 4:2 for example). In Revelation 4:4 there are twenty-four elders forming a “heavenly council” around the throne of God. There are a few examples in Acts of the word referring to early Christian leaders (Acts 11:30; 14:23; 15:2-6; 16:4), although with the exception of 14:23 these all Jewish congregations so they may have taken over the leadership structure of the synagogue; this may also be true for Paul’s congregations.
While it is always possible this Elder is not the elderly Apostle John, there is little in 1-3 John to suggest the Apostle is not the authority behind the letters. Eusebius cites Clement of Alexandria, John was based on Ephesus but traveled throughout the region appointing bishops and dealing with church issues. Assuming the writer is the same as 1 John 1:1-4, the claim to be an eyewitness of Jesus is more or less a claim to apostolic status.
Is the “chosen lady” an actual person? The word John uses in verse one (κυρία) refers to a woman of special status. Some suggest this is a woman named Eklete (Greek word for chosen) who led the church to which John is writing. The “mistress of the house” would refer to the woman in charge of assigning work to all the household slaves and managing household affairs but in the LXX the mistress (κυρία) is not a slave (BDAG). The woman may be like Lydia or Phoebe, a wealthy woman who hosted the church in her home and was the patroness of a local congregation. In support of this view, see this post by Marg Mowczko arguing the Lady is the leader of a house church. Here are Paul Anderson’s comments on the Lady as a real church leader.
However, the consensus answer is the Chosen Lady is a metaphor for a house church rather than a literal woman who led a church. The Greek word translated church, gathering, or congregation (ἐκκλησία) is feminine. Using a metonym for the church like this is unusual in the New Testament, but in the ancient world cities were often described in female terms (Jerusalem as a woman in Isaiah), Paul used a similar metaphor in Ephesians 5:25 and 2 Corinthians 11:2. The children (and her sister’s children in v. 13) would then refer to the members of two separate congregations.
Although I agree with Marg that Phoebe was deacon or minister of her church (Rom. 16:1–2) and that Junia (Rom 16:7) and Nympha (Col 4:15) are examples of women in leadership in local churches, I find it simpler to understand this particular chosen Lady as a reference to the church.
It is not clear why John used these metaphors to hide his identity and that of the congregation. 1 Peter 5:13 uses Babylon as a metonym for Rome (both were arrogant world empires which had destroyed Jerusalem and the Temple). It may be the case the writer was masking his identity and the location of the congregation to avoid drawing attention of the authorities.