It seems common in Christian communities (and in questions on this site) for people to say they have been saved. What does someone mean by this when they say it? How would I know that I have been saved and is there a particular moment of saving I should be able to identify?
In order for the language of being "saved" to make sense one has to understand what we are being saved FROM. There has to be a before and after scenario. We are saved from the wrath of God. Those without faith in Christ are described as children of wrath.
Consider this passage that uses the language of being "saved" including some details on our original position before God, the change that happens, and our final standing after salvation as well as the key operative bits on how that actually happens:
Ephesians 2:1-9 (ESV)
1 And you were dead in the trespasses and sins 2 in which you once walked, following the course of this world, following the prince of the power of the air, the spirit that is now at work in the sons of disobedience— 3 among whom we all once lived in the passions of our flesh, carrying out the desires of the body and the mind, and were by nature children of wrath, like the rest of mankind. 4 But God, being rich in mercy, because of the great love with which he loved us, 5 even when we were dead in our trespasses, made us alive together with Christ—by grace you have been saved—6 and raised us up with him and seated us with him in the heavenly places in Christ Jesus, 7 so that in the coming ages he might show the immeasurable riches of his grace in kindness toward us in Christ Jesus. 8 For by grace you have been saved through faith. And this is not your own doing; it is the gift of God, 9 not a result of works, so that no one may boast.
That wrath was meted out on Christ (Isaiah 53:5) and by bearing this punishment and separation from the Father, He claimed the right to make us Sons. He transferred us from the domain of darkness into the light.
Colossians 1:1-14 (ESV)
11 May you be strengthened with all power, according to his glorious might, for all endurance and patience with joy, 12 giving thanks to the Father, who has qualified you to share in the inheritance of the saints in light. 13 He has delivered us from the domain of darkness and transferred us to the kingdom of his beloved Son, 14 in whom we have redemption, the forgiveness of sins.
Returning to the language of Ephesians above, "being saved" means being made alive, to be raised up with Christ. A few verses after the Colossians passage above it talks about how God reconciled us to Himself through Christ.
To "be saved" thus means to be rescued from the wrath of God, made alive in Christ and reconciled to the Father. The pre-salvation scenario thus has us as an object of God's wrath, the post-salvation scenario has us as objects of His affections and co-heirs with Christ.
Jesus died that we may be saved from sin. That salvation has several aspects.
Saved from the penalty of sin
We have broken God’s laws, so we should receive the death penalty. But when Jesus died on the cross, God placed on him the blame for all the sins of the world. Jesus bore the penalty for our past sins so that we would not have to. Once removed, God cannot put the penalty for those particular sins back onto us.
Saved from the power of sin
In the past, I sinned because it was my nature to do so. It was normal to have nothing to do with God all day long. My mind was set upon what I could see, feel, taste, touch, and hear in the world around me. I was moved by those things. Saved from the power of sin, I can now be moved by spiritual things, such as the love of God.
Saved from the corruption of sin
We were made in the image of God. But sin had corrupted that image. When I believed in Jesus Christ, God put the Spirit of Christ in me. Spiritually, my sinful nature went into the tomb with Jesus, and my new nature rose from the dead with Jesus. Now over time I can become more like Jesus as he lives his life through me. This is the process of sanctification, where we are being saved – salvation is an ongoing process. You can reverse this process by various grievous choices.
Saved from the presence of sin
When we die and are resurrected, we will be saved from the presence of sin. Sin will not inhabit our resurrected bodies. Sin will have no place in the kingdom of God to come.
The Roman Road is a good "path" of sorts for receiving salvation, but it is Romans 10 that is especially appropriate for this question.
Romans 10:9-10 (NLT)
9 If you confess with your mouth that Jesus is Lord and believe in your heart that God raised him from the dead, you will be saved. 10 For it is by believing in your heart that you are made right with God, and it is by confessing with your mouth that you are saved.
Romans 10:13 (NLT)
13 For “Everyone who calls on the name of the Lord will be saved.”
I can't say it any better than these verses.
As for the second part of your question,
". . . is there a particular moment of saving I should be able to identify?"
There are some Christians, at least in my sphere of influence, who believe in a "point in time" salvation. Truth be told, many Christians can and do point to a specific day when they were saved. For them, the circumstances were such that the day of their salvation is indelibly etched in their minds.
This point-in-time salvation is legitimate, powerful, and quite persuasive, and the church and the world benefit from such testimonies. I call people in this category "trophies of [God's] grace," and for good reason. For them, their "before" (their BC, before Christ) and their "after" (their AD, after deliverance) are like night and day. God's unmerited favor shone into their lives in dramatic ways.
Some former drug addicts, for example, testify to being delivered from their addiction virtually instantly. Some atheists or agnostics testify to a day and time when the light finally dawned on them and they stepped from unbelief into belief. The Apostle Paul, of course, had a dramatic conversion, complete with a literal voice from God asking him why he was persecuting Jesus by persecuting Jesus' church, a literally blinding light, and a miraculous healing of that blindness just days after it occurred with the help of a man to whom God appeared in a vision (see Acts 9, 22, and 26:9-18).
On the other hand, some Christians in my tiny sphere of influence do not put much stock in point-in-time conversions, and for various reasons. That is not to say they do not believe in such conversions; it is to say they are more flexible in how they look at the whole conversion process. To them, conversion is just as likely to occur over time in stages, with one link in the chain being joined to the next link in the chain until the chain finally reaches from heaven to earth, as it were.
For such folks, if you were to ask them "When were you saved?" they would register a quizzical look on their faces and with a shrug of their shoulders say,
"Gee, I don't know. I just know I'm saved. As to the 'when' of it all, I'm drawing a blank. All I know is that my being saved was a gradual process, and I cannot point to a specific time and place and say 'That's where and when I was saved.'"
As to where I stand in this matter, I can pinpoint fairly accurately the when and where of my conversion. It was not a dramatic conversion, by any means, because I was only seven years old at the time! Years later, as a teen, I was struggling with what is commonly called "assurance of salvation," or more accurately "a lack of assurance of salvation," and I told God, again, I was really serious about being saved.
Looking back, however, who is to say which experience was the one that clinched things for me? All I know is at the age of 63, almost 56 years after my childhood conversion (if indeed it was that), I am saved and have full assurance of it.
In conclusion, while all Christians can have full assurance of salvation (see 1 John 5:13), can anyone truly KNOW she or he is saved? If so, then where does faith enter the picture? Without getting into the intricacies of epistemology (loosely, the study of the nature of knowledge and how we know what we know), I suggest we can never, this side of heaven, know we are saved. We can, however, through the eyes of faith say with the Apostle Paul,
"I know whom I have believed, and I am persuaded that He is able to keep what I have committed unto Him until the day [of His appearing]" (2 Tim 1:12)
Ultimately, I suggest, our assurance of salvation rests on the unchanging character of the One who saved us, and when you KNOW Him, you simply know you are saved. Will occasional doubts creep in from time to time? Of course. When they do, we must then fall back on the promises of God's word, such as
"'My sheep hear My voice, and I know them, and they follow me; and I give eternal life to them, and they will never perish; and no one will snatch them out of My hand. My Father, who has given them to Me, is greater than all; and no one is able to snatch them out of the Father's hand'" (John 10:27-29).
After we all die and we all stand before Christ to be judged for what we have done, some people will fail the test and be sent to Hell. Others will have succeeded in this life, be declared righteous and right with God and go on to live with Christ for the rest of eternity.
To be Saved by the blood of Christ means that you are free from the Wrath of God's Justice
In order to be saved you must: Does believing in Jesus Christ save me, or do I have to do something more?
To know that you are saved, look here: How can I be sure someone I know is going to Heaven?
I think it important to understand the status quo; wickedness.
11 Say unto them, as I live, saith the Lord God, I have no pleasure in the death of the wicked; but that the wicked turn from his way and live: turn ye, turn ye from your evil ways; why will ye die, O house of Israel?
12 Therefore, thou son of man, say until the children of thy people, The righteousness of the righteous shall not deliver him in the day of his transgression: as for the wickedness of the wicked, he shall not fall thereby in the day that he turneth from his wickedness: neither shall the righteous be able to live for his righteousness in the day that he sinneth.
Repentance. Isaiah 55:7 (KJV)
Let the wicked forsake his way, and the unrighteous man his thoughts: and let him return unto the Lord, and he will have mercy upon him: and to our God, for he will abundantly pardon
To be saved, is to be born of water and of the Spirit.
Jesus answered and said unto him, Verily, verily, I say unto thee, Except a man be born of water and of the Spirit, he cannot enter into the kingdom of God.
And said, Verily I say unto you, Except ye be converted, and become as little children, ye shall not enter into the kingdom of heaven.
What does someone mean when they say they have been saved? Hopefully, they mean what they say.
Not every one that saith unto me, Lord, Lord, shall enter into the kingdom of heaven; but he that doeth the will of my Father which is in heaven.
How would I know that I have been saved?
Then said I , Woe is me! For I am undone; because I am a man of unclean lips, and I dwell in the midst of a people of unclean lips: for mine eyes have seen the King , the Lord of hosts.
You will be sorrowful and really understand just how insignificant and powerless you are in helping yourself. A humbling experience.
Is there a particular moment I should be able to identify? Yes
When you turn your life around 180 degrees, you no longer do what you did or think what you thought in the past. If you have truly surrendered to the Holy Spirit, you are no longer in charge. It is not you who directs your actions, it is the work of the Holy Spirit.