Perseverance of the Saints

Perseverance may be defined as that continuous operation of the Holy Spirit in the believer, by which the work of divine grace that is begun in the heart is continued and brought to competition. It is because God never forsakes His work that believers continue to stand to the very end.
Louis Berkhof from Systematic Theology (pg. 546)

The doctrine of the perseverance of the saints does not maintain that all those who profess the Christian faith are certain of heaven. It is saints - those who are set apart by the Spirit - who persevere to the end. It is believers - those who are given true, living faith in Christ - who are secure and safe in Him. Many who profess to believe fall away, but they do not fall from grace for they were never in grace. True believers do fall into temptations, and they do commit grievous sins, but these sins do not cause them to lose their salvation or separate them from Christ. The Westminster Confession of Faith gives the following statement on this doctrine:
    They whom God hath accepted in His Beloved, effectually called and sanctified by His Spirit, can neither     totally nor finally fall away from the state of grace: but shall certainly persevere therein to the end, and     be eternally saved.
Boettner is certainly correct in asserting that
    This doctrine does not stand alone but is a necessary part of the Calvinistic system of theology. The     doctrines of Election and Efficacious Grace logically imply the certain salvation of those who receive     these blessings. If God has chosen men absolutely and unconditionally to eternal life, and if His Spirit     effectively applies to them the benefits of redemption, the inescapable conclusion is that these persons     shall be saved. The following verses show that God's people are given eternal life the moment they     believe. they are kept by God's power through faith and nothing can separate them from His love. They     have been sealed with the Holy Spirit who has been given as the guarantee of their salvation, and     they are thus assured of an eternal inheritance.
David Steele & Curtis C Thomas from The Five Points of Calvinism (pg. 56)

Perseverance of the saints is the biblical doctrine that God infallibly preserves in faith all of those he has given to the Son (John 6:37, 39, 44, 63-65) so that they are never lost. It maintains that none who are truly redeemed by Christ can be condemned for their sins or finally fall away from the faith. For as the apostle Paul states in Philippians 1:6, “I am sure of this, that he who began a good work in you will bring it to completion at the day of Jesus Christ.”
Monergism.com