Conversion is a turning from evil to God. God converts (Acts 21:19) the unsaved into the saved, from the unregenerate to the regenerate. It is produced through the preaching of the gospel (Romans 10:14; 1 Corinthians 15:1-4) and results in repentance (Acts 26:20) and a new creation (2 Corinthians 5:17). The fruit of conversion are listed in Galatians 5:22-23.
To be justified is to declared legally righteous. It is a divine act where God declares the sinner to be innocent of their sins. It is not that the sinner is now sinless but that they're "declared" sinless. The sinner is not made righteous in that their soul is changed or that their soul is infused with God's grace. Instead, justification is a legal act of imputing the righteousness of Christ to the believer (Romans 4:11, Philippians 3:9). This justification is based on the shed blood of Jesus, " . . . having now been justified by His blood . . . " (Romans 5:9). When God sees the believer, He sees us through the sacrifice of Jesus and "sees" us without sin. This declaration of innocence is not without cost, for it required the satisfaction of God's Law, " . . . without shedding of blood there is no forgiveness," (Hebrews 9:22). By the sacrifice of Jesus and in the "one act of righteousness, justification is given to all that believe." (Romans 5:18). In justification, the justice of God fell upon Himself (Jesus). We receive mercy. We are not judged according to our sins, and grace is shed upon us. We receive eternal life. This justification is a gift of grace (Romans 3:24) and by faith (Romans 3:28) because Jesus bore our guilt (Isaiah 53:12).
To sanctify means to be set apart for a holy use. God has set us apart for the purpose of sanctification not impurity (1 Thessalonians 4:7), and being such we are called to do good works (Ephesians 2:10). The believer is to sanctify Christ as Lord in their heart (1 Peter 3:15). God sanctified Israel as His own special nation (Ezekiel 27:28). People can be sanctified (Exodus 19:10, 14) and so can a mountain (Exodus 19:23) as can the Sabbath day (Genesis 2:3), and every created thing is sanctified through the word of God and prayer (1 Timothy 4:4-5).
Sanctification follows justification. In justification our sins are completely forgiven in Christ. Sanctification is the process by which the Holy Spirit makes us more like Christ in all that we do, think, and desire. True sanctification is impossible apart from the atoning work of Christ on the cross because only after our sins are forgiven can we begin to lead a holy life.