This is a question that is even asked in the Bible (Acts 16:31, see Matthew 19:16). There is no more important question. Many answers have been given, often from personal experience, but only that which is true to the Bible is to be trusted.
If you need to be saved, then there is something you need to be saved from. There is no doubt that the great problem from which we all need to be saved, is SIN. Jesus came to save His people from their sin (Matthew 1:21, see Luke 1:77). This not how it is often taught today, where the emphasis is on the need for healing, or for prosperity, or the particular need you feel. How serious is this sin? Is it like a flu from which we should recover? Or is it like terminal disease about which we can do nothing? So consider that sin is not just our actions against God’s law, but it comes from a sinful heart, as water comes from a spring – so Jesus clearly taught (Matthew 12:34, 15:18-19). And because sin controls our hearts, it holds us captive not able to give genuine obedience to God’s commands (Romans 8:7). All the Roman Christians were such slaves to sin before they were saved (see 6:15-23). A slave cannot free himself but needs someone more powerful than his master to free him. One who is dead in sin (Ephesians 2:1,5) needs someone to give new life. When you have begun to see sin as this great problem then you will seek to find the only salvation that can really deliver you from it. The medicine is always prescribed according to the nature of the disease.
Salvation is through faith alone in Jesus Christ alone: John 3:16-18, Acts 16:31, Romans 10:13, Ephesians 2:8. True saving faith is not a work, and so it is often contrasted with works (see Romans 3:28, 4:5-6, Ephesians 2:8-9). But sadly faith often is made a work, as when it is emphasized that it is a person’s free-will decision, or when it is coming to the front to repeat the so-called ‘sinner’s prayer’. It becomes a work because it is what the sinner trusts in as the assurance of salvation. You are told, ‘Don’t doubt, you have repeated the prayer.’ However, faith is not trust in anything you have done or said; but it is trust in God, what He has done in Christ and what he has said about it in Word. What is it to have the faith which saves? You first hear the good news of Jesus Christ that He is willing to save you from your sins because He has said, “Come to Me, all who labour and are heavy laden,…” (Matthew 11:28). Then you hear that He is able to save you because He died on the cross for sinners, bearing the punishment they deserved for their sins (1 Corinthians 15:3). Finally, you reject any trust in yourself, who you are by your upbringing, anything that you have done, and put all your trust in Jesus Christ to save you and bring you to God. This is what Paul had to do to be saved – read Philippians 3:4-9. See Biblical Answers No. 5 on False Faith.
That God gets all the glory shows that it is His work and not ours. When you present yourself at the hospital to be operated, afterwards you only praise the surgeon in whom you put your trust. After Paul has laid bare the awful condition of sinners (Ephesians 2:1-3), he then shows the way of salvation, starting with the words, “But God” (v. 4, see also Titus 3:4). Our contribution is the sin; it is God who saves. Even the faith we have in Christ is the gift of God. This is why, when Paul has fully explained God’s salvation he ends with, “For from him and through him and to him are all things. To him be glory forever. Amen” (Romans 11:36). Never are Christians congratulated for making their decision to follow Christ, but it is always thanksgiving to God, for He alone has done it, by His grace alone. Having told the Romans that they had been helpless slaves to sin, it is God he thanks for saving them from such a condition (Romans 6:16-18). In many of his letters Paul gives thanks to God for the saving grace they have experienced (e.g. 1 Corinthians 1:4, Colossians 1:3, 1 Thessalonians 1:2). Ask yourself, why is it you who God has saved and not some other member of your family? Is it because you are better, or wiser, or stronger? You know that is not true, which is why you yourself thank God that you are a Christian!
(1) Won’t we be saved by works on the Judgement Day? It is true that we will be judged by our works on the last day (see Romans 2:6, Revelation 20:12-13). This does not contradict salvation by faith alone. The Bible is very clear that genuine saving faith always shows itself in the fruit of works (Galatians 5:6, Ephesians 2:8-10, James 2:14-26). Works are the evidence of faith. The fruit is not the tree, but it is the evidence the tree is alive.
(2) Don’t we need to repent to be saved? Yes, there is no forgiveness of sin without repentance (Acts 2:38). Repentance is inseparably joined to faith (as in Mark 1:15, Acts 20:21). Why does anyone turn to Christ to save them? Because we know our sins will bring us to judgement, and we are turning away from them to Christ not just to be forgiven but to lead a new life of pleasing God, and that is repentance. Repentance is not something the sinner does that God rewards; it is rather a humble acknowledgement that we are the sinner God says we are! To admit guilt does not merit acquittal!
(3) Is it possible to know for sure that you are saved? Yes! Paul refers to the Ephesians as already having been saved (Ephesians 2:8). John wrote his first letter to give such assurance to those who believe (5:13). It is because salvation is through faith in Christ that we may rightly have assurance. If it in any depends upon our works then assurance is impossible because we do not know what we might do in the future. This is why there is no assurance in the cults. But those why are justified by faith are already rejoicing in hope of the glory of God (Romans 5:1-2)! Our faith is in Christ. What has He done? He has died for our sins, for all of them, not just past sins. So “There is therefore now no condemnation for those who are in Christ Jesus” (Romans 8:1). On the Day of Judgement Christ will be there, as the Lamb who was slain, and that is sufficient to bring us to glory. Praise the Lord!