What It Means To Be Anointed

A) To be consecrated:

To be set apart or consecrate: An anointed person is a consecrated person. In the Old Testement, priests and kings were anointed with for their consecration to the service of The Lord.

 Exodus 29:29

And the holy garments of Aaron shall be his sons 'after him, to be anointed therein, and to be consecrated in them.

 Leviticus 4:3

If the priest that is anointed do sin according to the sin of the people; then let him bring for his sin, which he hath sinned, a young bullock without blemish unto the Lord for a sin offering. 

1 Samuel 2:10

The adversaries of the Lord shall be broken to pieces; out of heaven shall he thunder upon them: the Lord shall judge the ends of the earth; and he shall give strength unto his king, and exalt the horn of his anointed.(Consecrated one)

 Isaiah 61:1

 The Spirit of the Lord God is upon me; because the Lord hath anointed (consecrated) me to preach good tidings unto the meek; he hath sent me to bind up the brokenhearted, to proclaim liberty to the captives, and the opening of the prison to them that are bound;

B) To be marked or identified:

2 Kings 9:3 Then take the box of oil, and pour it on his head, and say, Thus saith the Lord, I have anointed (marked) thee king over Israel. Then open the door, and flee, and tarry not.

C) To be made prosperous:

Psalms 23:5

Thou preparest a table before me in the presence of mine enemies: thou anointest my head with oil; my cup runneth over

D)  To be consecrated into an office:

Acts 10:38 How God anointed Jesus of Nazareth with the Holy Ghost and with power:who went about doing good, and healing all that were oppressed of the devil; for God was with him.

When Jesus was anointed by the Holy Spirit, he was consecrated into his Messianic office and given the gifts that go with that office

The Greek word used her is "Chrio" to smear or rub with oil, i. e. (by implication) to consecrate to an office or religious service:— anoint, consecrate.