Sanctification (hagiasmos) and Sanctify (hagiazo)

Hagiasmos occurs ten times in the NT, translated in the KJV as sanctification five times and holiness five times. The verb Hagiazo occurs 29 times,  26 times as sanctify and twice as hallowed ("hallowed be Thy name"). It means 'to set apart'. But how and why something is being set apart depends on the context. The root word is hagios, translated 161 times as "holy", 61 times as "saints" and four times as "Holy One".

Hagiasmos (sanctification)
The practical necessity of sanctification:

..for as ye have yielded your members servants to uncleanness and to iniquity unto iniquity; even so now yield your members servants to righteousness unto holiness. For when ye were the servants of sin, ye were free from righteousness. What fruit had ye then in those things whereof ye are now ashamed? for the end of those things [is] death. But now being made free from sin, and become servants to God, ye have your fruit unto holiness, and the end everlasting life.  
Rom. 6:19-22  [the fruit of righteousness (love, peace and joy) only grows in the heart whose 'garden' is set apart from sin. Hagiasmos in this 'practical holiness' context is also emphasized in 1Thess 4:3-7]

A big picture look at sanctification:
Elect according to the foreknowledge of God the Father, through sanctification
of the Spirit, unto obedience and sprinkling of the blood of Jesus Christ: Grace unto you, and peace, be multiplied. 1Peter 1:2  [the Holy Spirit's work in a believer is to set us apart unto the grace of God, which he does by building us up in God's word and producing the fruits of righteousness]

Hagiazo (sanctify)
The setting apart of Jesus from the glory of heaven to the work of salvation on earth:
Say ye of him, whom the Father hath sanctified, and sent into the world, Thou blasphemest; because I said, I am the Son of God?  John 10:36

The setting apart of sinners by the gospel call and choosing of the God from death and destruction to life and blessing in Christ:
And now, brethren, I commend you to God, and to the word of his grace, which is able to build you up, and to give you an inheritance among all them which are sanctified. Acts 20:32

That I should be the minister of Jesus Christ to the Gentiles, ministering the gospel of God, that the offering up of the Gentiles might be acceptable, being sanctified by the Holy Ghost. Rom.15:16

But we are bound to give thanks always to God for you, brethren beloved of the Lord, because God hath from the beginning chosen you to salvation through sanctification of the Spirit and belief of the truth: 2Thess. 2:13    also Acts 26:18, Jude 1:1, 1Cor.1:2. 1 Cor.6:11, Heb. 2;11, 1Thes. 5:23

The setting apart of sinners by the blood of Christ from sin and judgment to cleanliness and service:
For if the blood of bulls and of goats, and the ashes of an heifer sprinkling the unclean, sanctifieth to the purifying of the flesh: How much more shall the blood of Christ, who through the eternal Spirit offered himself without spot to God, purge your conscience from dead works to serve the living God?
Heb 9:13-14  also Heb 10:10, Heb 13:12, Eph. 5:26

The everyday setting apart of a believers walk from the ways of the world to the ways of God:
But sanctify the Lord God in your hearts: and [be] ready always to [give] an answer to every man that asketh you a reason of the hope that is in you with meekness and fear.

If a man therefore purge himself from these, he shall be a vessel unto honour, sanctified, and meet for the master's use, [and] prepared unto every good work. 2Tim.2:21