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. 32:1717And the work of righteousness shall be peace; and the effect of righteousness quietness and assurance for ever. (Isaiah 32:17)): whereas in their disobedience they should fear day and night and have no assurance of their life (Deut. 28:6666And thy life shall hang in doubt before thee; and thou shalt fear day and night, and shalt have none assurance of thy life: (Deuteronomy 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. 1:55For our gospel came not unto you in word only, but also in power, and in the Holy Ghost, and in much assurance; as ye know what manner of men we were among you for your sake. (1 Thessalonians 1:5)). We also meet with:
2. the full assurance of hope (Heb. 6:1111And we desire that every one of you do show the same diligence to the full assurance of hope unto the end: (Hebrews 6:11)), issuing in continued diligence of the saints in their work and labor of love: and

From Anstey’s Doctrinal Definitions:

This term is used in two main ways in the New Testament:
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. 10:2222Let us draw near with a true heart in full assurance of faith, having our hearts sprinkled from an evil conscience, and our bodies washed with pure water. (Hebrews 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 CROSSScripture says that when the Lord Jesus Christ was on the cross, God “laid on Him” the sins of all who would believe (Isa. 53:66All we like sheep have gone astray; we have turned every one to his own way; and the Lord hath laid on him the iniquity of us all. (Isaiah 53:6)). During the three hours of darkness Christ “bore” the judgment for all those sins (Heb. 9:2828So Christ was once offered to bear the sins of many; and unto them that look for him shall he appear the second time without sin unto salvation. (Hebrews 9:28); 1 Peter 2:2424Who his own self bare our sins in his own body on the tree, that we, being dead to sins, should live unto righteousness: by whose stripes ye were healed. (1 Peter 2:24)). Thus, the price of the believer’s sins has been fully paid through Christ’s “finished” work (John 19:3030When Jesus therefore had received the vinegar, he said, It is finished: and he bowed his head, and gave up the ghost. (John 19:30)). The claims of divine justice have been satisfied (Psa. 85:1010Mercy and truth are met together; righteousness and peace have kissed each other. (Psalm 85:10); Isa. 53:1010Yet it pleased the Lord to bruise him; he hath put him to grief: when thou shalt make his soul an offering for sin, he shall see his seed, he shall prolong his days, and the pleasure of the Lord shall prosper in his hand. (Isaiah 53:10)), and God has been “glorified” in the whole matter (John 12:27-28; 13:31-3227Now is my soul troubled; and what shall I say? Father, save me from this hour: but for this cause came I unto this hour. 28Father, glorify thy name. Then came there a voice from heaven, saying, I have both glorified it, and will glorify it again. (John 12:27‑28)
31Therefore, when he was gone out, Jesus said, Now is the Son of man glorified, and God is glorified in him. 32If God be glorified in him, God shall also glorify him in himself, and shall straightway glorify him. (John 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. 4:24-2524But for us also, to whom it shall be imputed, if we believe on him that raised up Jesus our Lord from the dead; 25Who was delivered for our offences, and was raised again for our justification. (Romans 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 1:2121Who by him do believe in God, that raised him up from the dead, and gave him glory; that your faith and hope might be in God. (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. 9:1414How 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? (Hebrews 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. 3:1616And without controversy great is the mystery of godliness: God was manifest in the flesh, justified in the Spirit, seen of angels, preached unto the Gentiles, believed on in the world, received up into glory. (1 Timothy 3:16)). God has seated Him in the highest place in heaven—at His own right hand (Eph. 1:20-2120Which he wrought in Christ, when he raised him from the dead, and set him at his own right hand in the heavenly places, 21Far above all principality, and power, and might, and dominion, and every name that is named, not only in this world, but also in that which is to come: (Ephesians 1:20‑21); Phil. 2:9-119Wherefore God also hath highly exalted him, and given him a name which is above every name: 10That at the name of Jesus every knee should bow, of things in heaven, and things in earth, and things under the earth; 11And that every tongue should confess that Jesus Christ is Lord, to the glory of God the Father. (Philippians 2:9‑11); Col. 3:11If ye then be risen with Christ, seek those things which are above, where Christ sitteth on the right hand of God. (Colossians 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. 1:1313Thou art of purer eyes than to behold evil, and canst not look on iniquity: wherefore lookest thou upon them that deal treacherously, and holdest thy tongue when the wicked devoureth the man that is more righteous than he? (Habakkuk 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. 23:1919God is not a man, that he should lie; neither the son of man, that he should repent: hath he said, and shall he not do it? or hath he spoken, and shall he not make it good? (Numbers 23:19); Heb. 6:17-1817Wherein God, willing more abundantly to show unto the heirs of promise the immutability of his counsel, confirmed it by an oath: 18That by two immutable things, in which it was impossible for God to lie, we might have a strong consolation, who have fled for refuge to lay hold upon the hope set before us: (Hebrews 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. 12:2323To the general assembly and church of the firstborn, which are written in heaven, and to God the Judge of all, and to the spirits of just men made perfect, (Hebrews 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 5:2424Verily, 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: