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:
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.
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.
As for the second part of your question,
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,
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,
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
I think it important to understand the status quo; wickedness.
Repentance. Isaiah 55:7 (KJV)
To be saved, is to be born of water and of the Spirit.
What does someone mean when they say they have been saved? Hopefully, they mean what they say.
How would I know that I have been saved?
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.
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?
The Roman Road is a good "path" of sorts for receiving salvation, but it is Romans 10 that is especially appropriate for this question.
I can't say it any better than these verses.
Jesus died that we may be saved from sin. That salvation has several aspects.
Saved from the penalty of sin
Saved from the power of sin
Saved from the corruption of sin
Saved from the presence of sin