But Daniel purposed in his heart that he would not defile himself with the portion of the king's meat, nor with the wine which he drank: therefore he requested of the prince of the eunuchs that he might not defile himself. Daniel 1:8
His was a biblical separatism because his standard was based on God's revelation in the Mosaic Law.
A modern example of personal separation could be the decision to decline invitations to parties where alcohol is served.
Such a decision might be made to circumvent temptation.
But put ye on the Lord Jesus Christ, and make not provision for the flesh, to fulfil the lusts thereof. Romans 13:14
...to avoid the "appearance of evil".
Abstain from all appearance of evil. 1 Thessalonians 5:22
...or simply to be consistent with a personal conviction.
One man esteemeth one day above another: another esteemeth every day alike. Let every man be fully persuaded in his own mind. Romans 14:5
The Bible clearly teaches that the child of God is to be separate from the world.
Be ye not unequally yoked together with unbelievers: for what fellowship hath righteousness with unrighteousness? and what communion hath light with darkness? And what concord hath Christ with Belial? or what part hath he that believeth with an infidel? And what agreement hath the temple of God with idols? for ye are the temple of the living God; as God hath said, I will dwell in them, and walk in them; and I will be their God, and they shall be my people. Wherefore come out from among them, and be ye separate, saith the Lord, and touch not the unclean thing; and I will receive you, 2 Corinthians 6:14-17
Wherefore gird up the loins of your mind, be sober, and hope to the end for the grace that is to be brought unto you at the revelation of Jesus Christ; as obedient children, not fashioning yourselves according to the former lusts in your ignorance: But as he which hath called you is holy, so be ye holy in all manner of conversation; 1 Peter 1:13-15
Corporate separation involves the decisions of the church concerning its ties to other organizations, based on their theology or practices. Separatism is implied in the very word church . The Greek word ekklesia means "a called-out assembly."
In Jesus' letter to the church of Pergamos, He warned against tolerating those who taught false doctrine.
But I have a few things against thee, because thou hast there them that hold the doctrine of Balaam, who taught Balac to cast a stumblingblock before the children of Israel, to eat things sacrificed unto idols, and to commit fornication. So hast thou also them that hold the doctrine of the Nicolaitans, which thing I hate. Revelation 2:14-15
The church was, and is, to be separate, breaking ties with heresy.
Biblical separation does not require Christians to have no contact with unbelievers or heretics. Like Jesus, we should befriend the sinner without partaking of the sin. Paul expresses a balanced view of separatism.
I wrote unto you in an epistle not to company with fornicators: yet not altogether with the fornicators of this world, or with the covetous, or extortioners, or with idolaters; for then must ye needs go out of the world. 1 Corinthians 5:9-10
In other words, we are in the world, but not of it.
The Pilgrim's Progress, by John Bunyan, provides a perfect example of this Biblical separation: