What I've had explained to me, and it sounds right in my heart, is simply this:
Take a look at this verse for a sec, real quick:
Matthew 7:22-23
Basically, what Jesus is saying here, is that there are a lot of people who are going to claim to know Him, they're going to say "Didn't we talk about You? Didn't we profess You? Didn't we do great things in Your name?"
And He is going to say "Depart from Me, I never knew you."
What that means, is that there are many... many... MANY people who will call themselves "Christians". They will tell you that they are saved, they will tell you that they know Christ, but yet they have loads of unrepentant sin in their lives. Part of coming to Christ, is repenting for your sin and what you've done.
Every single person is a sinner, and remains a sinner until they die. Those who are Saved, are those who realize they are dead in their sin, they realize there's nothing they can do to fix this, and they realize that Jesus Christ is our Lord and Saviour, and the only way out of this is to come to Him, and seek His divine love, mercy and grace.
In doing so, the person needs to realize that their sin will lead them to hell if they don't Repent. To Repent, you actually need to be sorry about you sins, you need to realize that you've done evil, you need to hate the evil things you've done in your life, but most importantly, one needs to make an effort to turn away from these sinful things in their lives. Once the person has done this, the person needs to ask for Forgiveness, they need to profess with their lips that Jesus is Lord, and ask Him to come into their lives and take control. That is when they become Born Again and Saved, when they receive the indwelling of the Holy Spirit.
Now, note earlier where I said "make an effort". Some sins are incredibly powerful and incredibly difficult to overcome. That is why the Holy Spirit justifies us (the act of being "saved") upon becoming Born Again, but yet Sanctifies us over time. This act of sanctification over time makes us more like Christ with each passing day, gradually erasing sin out of our lives and making us more holy and guiding us to do good rather than evil.
Those who will get told on the Last Day, "Depart from Me, I never knew you." are the people who claim to be Christians, they might have even said things like the Sinner's prayer, or did altar calls or even people who say all the words, do all the motions... but they leave out the "feel sorry for their sins" part and continue their in their sinful ways. They think they are saved, but they really aren't, because their prayers of repentance weren't from the heart, they weren't true, and they didn't truly believe in faith that they were dead in their sin, or that there's anything at all wrong with the way they are living their lives.
Therefore, they claim to know Christ... but Christ never knew them, because they never came to Christ, truly. They asked for Him to come into their lives, but yet they had no intention of doing what Christ instructed them to do (turning away from their sin).
Also, I would like to add that the word "iniquity" is a synonym for "sin", however it goes deeper than that. Usually when you see "iniquity" in the Bible, God is talking about a long-term pattern of sin. Someone who habitually and regularly sins without repentance.
People like these can still be saved, but they have to truly repent, and truly turn away from these sins in their lives. It could be anything -- sexual sin, greed, lack of charity, lack of brotherly love, whatever it is. Doesn't matter. He will forgive, but a person has to come to Him truly from the heart, and not just repeat some group of words without emotion.
EDIT: Also, take a look at these two verses:
Matthew 7:16-17
He says "you will know them by their fruits" as an evil tree can only produce evil fruit, a good tree can only produce good fruit. So, if a Christian is truly saved, if you look at his life, you will know it -- said Christian should show some evidence of changing, some evidence of turning away from sin, some evidence of doing things for Jesus and God. If they don't show any of these things whatsoever, then there's a chance they might not be saved.
We cannot judge either way, though. That isn't our place.
You can also use this on yourself -- do you see change in your own life? Do you show a pattern of turning away (or at the very least, trying to) from former sins, or are you still doing the things you've been doing even after your attempt to come to Christ?