Basically something that is pure is void of any admixture which could cause the slightest infection. A pure Christian is one who is void of any worldly or anti-Biblical teachings or practices which would cause them to have a divided allegiance between Christ and something else. Normally, that dichotomy would be between Christ and Satan. Whenever a Christian lives worldly or even partakes in the slightest fashion, they have become adulterated and have lost their purity until repentance and restoration takes place.

Pure In Heart
(Psa 24:3-4 KJV) Who shall ascend into the hill of the LORD? or who shall stand in his holy place? {4} He that hath clean hands, and a pure heart; who hath not lifted up his soul unto vanity, nor sworn deceitfully.
(Mat 5:8 KJV) Blessed are the pure in heart: for they shall see God.
Purity in Doctrine
(Job 11:4 KJV) For thou hast said, My doctrine is pure, and I am clean in thine eyes.
Purity in Motive in Prayer
(Job 16:17 KJV) Not for any injustice in mine hands: also my prayer is pure.
Purity in Word
(Psa 12:6 KJV) The words of the LORD are pure words: as silver tried in a furnace of earth, purified seven times.
(Prov 15:26 KJV) The thoughts of the wicked are an abomination to the LORD: but the words of the pure are pleasant words.
Purity Exudes Testimony
(Prov 20:11 KJV) Even a child is known by his doings, whether his work be pure, and whether it be right.
(Prov 21:8 KJV) The way of man is froward and strange: but as for the pure, his work is right.
Purity is to be dwelled upon
(Phil 4:8 KJV) Finally, brethren, whatsoever things are true, whatsoever things are honest, whatsoever things are just, whatsoever things are pure, whatsoever things are lovely, whatsoever things are of good report; if there be any virtue, and if there be any praise, think on these things.
Purity of Conscience
(1 Tim 3:9 KJV) Holding the mystery of the faith in a pure conscience.
(2 Tim 1:3 KJV) I thank God, whom I serve from my forefathers with pure conscience, that without ceasing I have remembrance of thee in my prayers night and day;
Purity of the Christian Walk
(1 Tim 5:22 KJV) Lay hands suddenly on no man, neither be partaker of other men's sins: keep thyself pure.
Purity of Religion (Christian Walk) is evidenced by good works
(James 1:27 KJV) Pure religion and undefiled before God and the Father is this, To visit the fatherless and widows in their affliction, and to keep himself unspotted from the world.
Purity of Mind
(2 Pet 3:1 KJV) This second epistle, beloved, I now write unto you; in both which I stir up your pure minds by way of remembrance: