Concise Bible Dictionary:

This word has in the Old Testament a different application from that which it has in the New Testament In the former it is “confidence or trust,” and agrees with the hopes of God’s earthly people in connection with the security in which Israel will dwell when restored to their land, when all their enemies shall have been put down by divine power: the effect of righteousness will be “quietness and assurance forever,” (Isa. 32:17): whereas in their disobedience they should fear day and night and have no assurance of their life (Deut. 28:66).
In the New Testament the Greek word πληροφορἰα implies “full assurance” and refers to eternal salvation. The gospel reaches a soul in power, and in the Holy Ghost and in “much full assurance” (1 Thess. 1:5). We also meet with:
1. the full assurance of faith (Heb. 10:22); the reception of God’s testimony respecting the work of Christ and the glory He now enjoys:
2. the full assurance of hope (Heb. 6:11), issuing in continued diligence of the saints in their work and labor of love: and
3. the full assurance of understanding, (Col. 2:2), for full knowledge in the mystery of God.

From Anstey’s Doctrinal Definitions:

This term is used in two main ways in the New Testament:
•  Firstly, it denotes the believer’s firm persuasion of his soul’s eternal salvation in Christ (1 Thess. 1:5; Heb. 6:11; 10:22; 1 John 3:19).
•  Secondly, it has to do with his conviction of the veracity of the truth which the Word of God teaches (Col. 2:2; 2 Tim. 3:14).
As to the side of things that pertains to the salvation of the soul, unfortunately there are many who, after having believed, have doubts as to whether they are truly saved. Some live in fear of being judged for their sins if they don’t carry on faithfully in the Christian path. This is not at all what God wants for His children. He wants us to have “full assurance” in regard to this important issue (Heb. 10:22), because if a person lacks it, he will be hindered in his spiritual growth.
The Christian’s assurance of salvation is based on knowledge and faith. Knowledge, in this respect, could be summarized in three things:
•  Knowing what happened at the cross.
•  Knowing what happened at the grave.
•  Knowing where Christ is now at the right hand of God.
God would have us to believe the testimony of His Word concerning these three things. The believer who understands and in simple faith accepts these great facts, will have solid assurance of his salvation, and peace in his soul. Under normal conditions, he will never get into doubt about it again.
1) KNOWING AND BELIEVING WHAT HAPPENED AT THE CROSS—Scripture says that when the Lord Jesus Christ was on the cross, God “laid on Him” the sins of all who would believe (Isa. 53:6). During the three hours of darkness Christ “bore” the judgment for all those sins (Heb. 9:28; 1 Peter 2:24). Thus, the price of the believer’s sins has been fully paid through Christ’s “finished” work (John 19:30). The claims of divine justice have been satisfied (Psa. 85:10; Isa. 53:10), and God has been “glorified” in the whole matter (John 12:27-28; 13:31-32).
The finished work of Christ has so completely satisfied divine justice, that if God were to come after the believer in judgment for his sins, then He would prove to be unjust, because the price has been fully paid. It would be demanding a second payment for their sins. It is something that God will never do.
2) KNOWING AND BELIEVING WHAT HAPPENED AT THE GRAVE—To put the question beyond any doubt, God raised the Lord Jesus from the dead (Rom. 4:24-25). This is significant. The resurrection of Christ is God's public statement that He has accepted Christ's work on the cross as a full payment for the believer’s sins. It is His “seal of approval” and His "Amen" to what Christ accomplished in His death (1 Peter 1:21). It is important to understand that assurance is not found in our acceptance of Christ's finished work, but in knowing that God has accepted it! The sacrifice of the Lord Jesus Christ was an offering that was not made to us, but to God (Heb. 9:14), and the proof that God has accepted it is in His raising Him from the dead.
Mr. G. Cutting had an illustration that emphasizes this point. He said that in the matter of an ordinary debt, the creditor is the only one who has the right to subscribe "PAID" on the account. It would be of no value whatever if the debtor were to do it, because he has no authority to declare it settled. The debtor doing it would not give satisfaction to the creditor, nor would it give peace to the debtor. The debtor's assurance can only be a consequence of knowing that the creditor is satisfied. Likewise, in connection with the debt of our sins, the only way in which we will get assurance that it has been paid is to see that it has been settled with God. We must see that God is completely satisfied with the payment that Christ made for us, and His raising Him from the dead proves His full satisfaction. Thus, the way to get assurance deep down in the bottom of our souls is to see that the whole question has been settled up at the top before God. If the payment for our sins is enough for God, it should be enough for us, and this should settle any doubt that we might have.
3) KNOWING WHERE CHRIST IS NOW AT GOD’S RIGHT HAND—Not only has Christ’s work of atonement been accepted by God, but Christ Himself has been accepted before God. This is witnessed in His ascension (1 Tim. 3:16). God has seated Him in the highest place in heaven—at His own right hand (Eph. 1:20-21; Phil. 2:9-11; Col. 3:1). He stands there now as a glorified Man with all the favour of God resting on Him. That God could receive Him into heaven is the clearest proof that our sins are gone! When the Lord was on the cross, He had our sins on Him, and He couldn’t have taken them with Him into heaven, for God’s holiness is such that He cannot have sin in His presence (Hab. 1:13). Thus, Christ’s ascension is a triumphant thing. It proves that our sins are gone!
Thus, full assurance of our salvation belongs to the person who believes what the Word of God says about these three things, for God cannot go back on His Word (Num. 23:19; Heb. 6:17-18). If God were to go back on His Word, He would have more to lose than us! We would lose our salvation, but He would lose His credibility as being a holy and just God—and essentially would cease to be God! We can be assured, therefore, that He will not go back on His Word.
Solid assurance, therefore, is not had by looking within ourselves for confirmation, but by believing what God has stated in His Word concerning our salvation in Christ. Suppose that a person has been convicted of a certain crime and was facing a prison sentence, but it was reported to him that he had been pardoned, because of certain extenuating circumstances surrounding the incident and the good grace of the judicial court. How would he know for sure that he was truly pardoned? Would it be by reading the feelings of his heart? Or would it be by listening to hearsay? Or would it be by reading the certified copy of the actual pardon signed by the supreme court judge? Obviously, it will be by reading the official document that declares his pardon. Similarly, if we want to be certain of the fact that we have been saved. Our feelings and wishes, and what people might say about it will not be the evidence that we’ll need to give us peace—for those sources don’t have the authority to pronounce on the matter. We must go to the Word of God and see what “God the Judge of all” says about it (Heb. 12:23). We can put our fullest confidence in His Word for He cannot lie, and that will be all we will need for solid assurance of our salvation. His Word says, “Verily, verily, I say unto you, He that heareth My Word, and believeth on Him that sent Me, hath everlasting life, and shall not come into condemnation; but is passed from death unto life” (John 5:24).
In view of giving the believer settled peace about his sins, God has gone to great lengths in His Word to show that the believer's sins are gone—and gone forever. He uses various figures and expressions to describe the blessedness of this great fact, so that there could be no legitimate question in the mind of the believer who accepts the testimony of God's Word. Some of the things that God has done with our sins are:
•  He has made purification for our sins (Heb. 1:3).
•  He removes our sins as far as the east is from the west (Psa. 103:12).
•  He blots our sins out (Isa. 44:22; Psa. 51:1).
•  He casts our sins behind His back (Isa. 38:17).
•  He casts them into the deepest sea (Mic. 7:19).
•  He takes our sins away (1 John 3:5).
•  He washes our sins away (Rev. 1:5).
•  He cleanses us from our sins (1 John 1:7).
•  He forgives our sins (Rom. 4:7; Eph. 1:7).
•  He remembers our sins no more (Heb. 10:17).