Several Greek words are translated anoint.  Most of them have the same root.  So, they have been all listed together.

Luke 4:18 – 19  (Quoted from Isaiah 61:1 – 2)  “The Spirit of the Lord is upon Me because He has anointed Me to preach the gospel to the poor; He has sent Me to proclaim liberty to the captives and recovery of sight to the blind, to set at liberty those who are oppressed; to proclaim the acceptable year of the Lord.”  This is what Jesus read in the synagogue in Nazareth.  Jesus concluded with “Today this Scripture is fulfilled in your hearing.”  The vast majority of ancient rabbis saw this passage to be Messianic, so the audience would have understood it in that connection.  By applying this passage to Himself, Jesus was making a clear claim to being the Messiah.  So, to be “anointed” by God represented God’s endorsement.  Kings and some prophets were anointed with oil as a symbol of their appointment.  This image is figurative.  God did not physically pour oil on Jesus.

Hebrews 1:8 – 9  (Quoted from Psalm 45:6 – 7) “But to the Son He says, ‘Your throne, O God, is forever and ever; a scepter of righteousness is the scepter of your kingdom.  You have loved righteousness and hated lawlessness; therefore God, Your God, has anointed You with the oil of gladness more than Your companions.’”  The companions are humans (2:11).  The oil of gladness is not that of appointment but that of a bridegroom, as in Matthew 9:15 or John 3:29.

Acts 4:27 – 28  “For truly against Your holy servant Jesus, whom you anointed, both Herod and Pontius Pilate with the Gentiles and the people of Israel were gathered together to do whatever Your hand and Your purpose determined before to be done.”  Jesus is called anointed by God, acknowledging His position.

Acts 10:38  “…how God anointed Jesus of Nazareth with the Holy Spirit and with power, who went about doing good and healing all who were oppressed by the devil, for God was with Him.”  Peter refers to Jesus as anointed, but not with oil, rather with the Spirit and power.

2 Corinthians 1:21 – 22  “Now He who establishes us with you in Christ and has anointed us is God who has also sealed us and given us the Spirit in our hearts as a guarantee.”  Paul calls himself, Silas, and Timothy anointed, referring to their place in bringing the gospel to Corinth by the power of God.  All Christians are said to be sealed with the Holy Spirit in Ephesians 1:13.  In 2 Corinthians, Paul is making the point that he and Silas and Timothy are anointed in the sense of being God’s representatives.

1 John 2:20  “But you have an anointing from the Holy One and you all know.”

1 John 2:27  “But the anointing which you have received from Him abides in you, and you do not need that anyone teaches you concerning all things”  This anointing for all Christians relates to understanding provided by the indwelling Spirit.

Revelation 3:18  “I counsel you to buy from Me gold refined in the fire, that you may be rich; and white garments that you may be clothed, that the shame of your nakedness may not be revealed; and anoint your eyes with eye salve, that you may see.”  The figure of speech is like John 9:41, “If you were blind, you would have no sin; but now you say, ‘We see.’  Therefore your sin remains.”  The Laodiceans could not see their true condition, although they claimed to be in need of nothing.

Note: the anointing in James 5:14 is a completely different word meaning “to oil.”

Common (over)applications:

Some groups apply Luke 4:18 – 19 (Isaiah 49:8 – 9) to all Christians, so people are still being anointed by God to preach the gospel to the poor and the other functions listed in that passage.  Certainly, each of these functions should still be accomplished by “ambassadors for Christ.”  The prophecy of Isaiah was said to be “fulfilled.”  So, Jesus was saying that God built an illustration in Israel centuries earlier concerning the rebuilding of a ruined Israel.  That physical illustration was being played out as the Messiah came to rebuild the Kingdom of God.  But it is a bit of a stretch to appropriate the anointing of Jesus as King with our anointing to understanding.

Charismatic groups use “anointing” to assign God’s endorsement to a person, a kind of authority.  The passages above are all there are, so the voluminous doctrines on the subject are largely the work of an overactive imagination.