One of the best explanations on the subject, which often experiences confusion:

John Owen:

“The anointing is clearly the Holy Spirit which all believers receive (Eph 3:16, Phil 1:19). The anointing of believers     therefore consists in the giving of the Holy Spirit to them, from and by Jesus Christ, and the One who anoints them is not the Holy Spirit, but Christ. The anointing, that is, the Holy Spirit, conveys spiritual teaching by illuminating the minds of believers to understand the mind of God and the mysteries of the gospel. By it believers know the truth and adhere firmly to it in love and obedience.

“…the anointing remains the spring from which the secret refreshment and support are ministered to them. And if they saw more clearly into the mysteries of the will, love and grace of God in Christ, as they should, they would have more of the consolation of knowing their acceptance with God, as being kings and priests to him, and have a refreshment which the world knows nothing of.”

                         -John Owen, The Spirit and the Church, pp. 162-3