The Bible uses the language of "anointing" to communicate this idea of spiritual authority. The anointing of the Holy Spirit is that ministry He does while indwelling the Christian. Such ministry results in Christians having empowerment and insight to function and serve the Lord. At least two types of anointing are spelled out in scripture.
The first is what I call an "outward" anointing. This has to do primarily with offices of every sort, whether family, govermental or within the church. God grants grace and ability to achieve that particular office. Clearly unbelievers or believers occupy positions or stations in families (parents) and in the culture (politics, doctors, police officers, teachers, etc). For example, in the Old Testament we see godly King David and a Pagan King named Cyrus both referred to as "God's anointed" (Psalm 2 and Isaiah 45). Only Christian men are specifically assigned by God to occupy the Pastoral/Elder and Deacon offices of Christ's church. (1 Timothy 3:1-14 and Titus 1) and believers in general are called by God with graces that accompany other areas such as teaching, administrating. All authority comes from God. Period.
The second type of anointing, what we will deem "inner anointing", refers to that spiritual authority that only comes through saving faith in Jesus Christ. 1 John 2:20 and 2:27 describes the indwelling Holy Spirit as being the Christian's "anointing", meaning the source of delegated authority from God to discern and live out the Christian life. 2 Corinthians 1:21-22 also describes this second type of anointing: "Now He who establishes us with you in Christ and anointed us is God, 22 who also sealed us and gave us the Spirit in our hearts as a pledge." Every Christian has the indwelling Spirit, and thus possess this inner anointing. Though all Christians have the inner anointing, each Christian will operate in different measure based upon how much they have taken in the scripture and have obediently yielded themselves to the Spirit's filling or influence. (Ephesians 5:18; Galatians 5:1,16). The anointing of the Holy Spirit is simply the empowerment and insight the Christian has from the indwelling Holy Spirit at work in and through their lives.
How so-called Christians are anointed by God to do extraordinary things
How can a Christian, working in an otherwise non-Christian environment, affect change? Certainly whatever station one is at in life, whether a parent, a doctor, a teacher, a laborer, a pastor or whatever the case may be, God has created the boundaries and abilities that come with each one. God has called and placed every believer where He wants them so they can be used by Him to impact others.
In understanding the nature of one's vocation to do a certain task (again we will tie this to the term "outward anointing"), the Christian uniquely has that second anointing, that "inner anointing" or indwelling Holy Spirit. The Holy Spirit indwells the child of God to compell them work hard and love their neighbor as themselves. So how do these two anointings work together?
Over time, a Christian who has regularly given themselves to the scriptures and prayer will to the unsuspecting co-worker gain respect. As the Christian learns their place and accepts what they are doing as a call from God, their ability to be used of God can only increase.
Closing Thoughts Just knowing that where you are is ordained by God (outward anointing) and that all that you need to do your job is supplied by the Holy Spirit of God indwelling you (i.e inward anointing) supplies the basis for Christian spiritual authority. Such authority must not be viewed as imposing one's will on someone else but rather serving the Lord by loving one's neighbor for Jesus sake and doing one's job for God's glory. As Paul writes in Romans 14:7-8 "For not one of us lives for himself, and not one dies for himself;8 for if we live, we live for the Lord, or if we die, we die for the Lord; therefore whether we live or die, we are the Lord’s."