This is a pretty big topic, but I'm gonna stick up a bit of a start to it here.
Grace is the big thing in the New Testament. You can't really read it without getting a glimpse of it. Jesus came to help the hated in society. He brought a message of new beginnings, of starting afresh. He was quite happy to spend time with the hated in society; prostitutes, tax collectors, the sick, the poor, the oppressed. He offered a complete wiping clean of sins. It was a bit radical. I mean, let's face it, if someone's wronged you you don't want some guy coming up and saying 'your sins are forgiven'. It upset quite a few of the Pharisees as well (the religious authorities of the time). They said that only God has the power to forgive sins. If you believe Jesus to be the son of God, that's not really a problem. But our very nature is conditioned not to expect free gifts, to look for obligations and strings.
A lot of churches do attach strings. They don't want certain types of people to come. My parents were asked to leave a church because they weren't the right sort of people (in a curious reversal of what you'd expect, it was because they were too posh). They expect a certain dress code. I went to a different church to normal the other week, along with most of my youth group, and we realised when we got there that everyone was dressed up nice and posh... Thankfully they were really cheerful folks and didn't seem to mind that we were all pretty casual. But you get the picture. Many people are put of churches because they think they have to conform to a certain mould, have to be 'fixed' before they go in. But Jesus said that it's the sick who need a doctor, not the healthy. You're not supposed to be sorted before you get to church.
Now, grace (after that rather lengthy introduction...). At the heart of grace is being given something you don't deserve. That something is eternal life. You don't deserve it. Nothing you can ever do would let you earn the right to eternal life. God realised this. That's why he sent Jesus--so we don't have to earn it. We can just receive it. You can't repay him. You can only say thank-you, and mean it.
Grace is what wipes your slate clean. But it's a once and for all wiping clean. Buster, our pastor, likes to ask people "are you a sinner?". The answer, by the way, is no (assuming you've accepted Jesus). Your slate has been wiped clean, and when you sin again, it just gets wiped straight away. It doesn't even stick for a second. When God looks at you, he doesn't see all the mess, all the broken promises, all the tries that ended in failure. He sees perfection, because he sees Jesus.
The response to grace is to say thank-you and accept it. To pick it up and run with it. It's like this: imagine someone gives you a mega nice dress/suit. You wear it, but when you wear it, you're careful with it, aren't you? You don't go out and start mucking it up. It's like that with grace. Even if you know it isn't going to stick to you, is it right to take advantage of that? Love God with all your heart, soul, and mind, and do whatever you want. Think about it. If you love God with all your heart, soul and mind, are you gonna start doing stuff that would upset him, stuff that breaks his heart? If you have a really close friend, you respect them, you don't trample all over their emotions, you don't ignore their opinions, you don't lie to them, hit them (at least, I would hope you don't intentionally do so...). That's why that statement holds true. It isn't a license to do whatever you want, it's a license to enjoy life with God's guidance.
Your parents have probably given you rules/gave you rules when you were younger. Don't go up there without me, don't climb that tree. These rules are not meant to stop yourself enjoying yourself, they're meant to keep you safe. I'm straying slightly away from the main theme, but never mind, I think this is still relevant.
Jeremiah prophesied: "I will put my law in their minds and write it on their hearts. I will be their God, and they will be my people." Basically: you don't need a written law any more. God's put it in you, instead of on paper. Instead of writing it on tablets of stone, it's inscribed into our very being, because Christ died for us and offers to come into us and brings that with him.
"Therefore, there is now no condemnation for those who are in Christ Jesus, because through Christ Jesus the law of the Spirit of life set me free from the law of sin and death. For what the law was powerless to do in that it was weakened by the sinful nature, God did by sending his own Son in the likeness of sinful man to be a sin offering." (Romans 8). What this means is that: instead of being under the Old Covenant, which brought death because we could never live up to it, God decided to do both sides of the deal, keeping the commands himself and then letting us reap the benefits. So don't let anyone (including yourself) condemn you for what you have done and have not done. Sin is no longer your master, you belong to Christ instead. And because of that, you have to recognise that nothing you can do will make God love you any less, and no amount of piety will make God love you any more. He can't possibly love you any more than he already does. Don't go feeling 'oh woe is me, I've sinned again'. Just move on. Let God direct your steps and help you not to fall into the trap again.
And remember this: God made you special and he loves you very very very much. More than you could possibly comprehend.