Where does the phrase appear?
Paul uses the phrase only once. He charges Timothy to act fairly when dealing with elders and accusations in the church, placing that charge before God, Christ Jesus, and the elect angels (1 Timothy 5:21). It is a serious setting, not a detailed lesson about the ranks or history of angels.
The word elect means chosen. Paul does not explain further, perhaps because Timothy already knew the language. That leaves room for careful confidence: these angels are God's chosen servants, but Scripture does not invite us to build an elaborate system from this single phrase.
I charge thee before God, and the Lord Jesus Christ, and the elect angels, that thou observe these things without preferring one before another, doing nothing by partiality.
Faithful angels in God's service
Many Christians understand the elect angels as the faithful angels, set apart for God and loyal to him. This fits the Bible's contrast between angels who sinned and are reserved for judgement, and the holy heavenly servants who remain in God's presence (2 Peter 2:4; Matthew 25:31).
Jesus describes the Son of Man coming in glory with his holy angels (Matthew 25:31). The description does not tell us every detail about their choosing, but it does show that holy angels belong with Christ and take part in God's final purposes.
What angels do
Angels are not independent spiritual powers. They are servants of God. Hebrews describes them as ministering spirits sent in service of those who will inherit salvation (Hebrews 1:14). That does not mean every person has a named guardian angel, but it does show God's active care for his people.
They also worship God and rejoice when a sinner repents (Luke 15:10). Their attention is never meant to turn us towards themselves. When John fell down before an angel, the angel stopped him and directed worship to God alone (Revelation 22:8-9).
Are they not all ministering spirits, sent forth to minister for them who shall be heirs of salvation?
What this means for us
The elect angels remind us that God's world is bigger than what we can see. Yet the Bible does not make angels the centre of faith. They serve the God who has made peace through Christ and is reconciling people to himself (Colossians 1:19-20).
So it is good to be curious, while staying close to what Scripture actually says. Read 1 Timothy 5:21 in context. Paul's practical concern is integrity, fairness, and living responsibly before God, Christ, and the heavenly witnesses.
And I John saw these things, and heard them. And when I had heard and seen, I fell down to worship before the feet of the angel which shewed me these things. Then saith he unto me, See thou do it not: for I am thy fellowservant, and of thy brethren the prophets, and of them which keep the sayings of this book: worship God.