Knowing that various Christian traditions understand the interaction of sin and grace in a believer from different perspectives, here is how I piece together the biblical evidence (by the way that’s what theology is: a piecing together, weighing and organizing of what the Bible teaches):
Our Natural State: Total Depravity. As we are naturally born, our human nature is thoroughly infected with sin (Psalm 139, Jeremiah 17:9, etc). This does not mean that we are born totally evil. Obviously unbelievers are capable of doing good things. It does mean, however, that we are affected by sin in every component of who we are (body, emotions, mind, will, etc). The collective term in the scriptures for these components affected by sin is flesh. This concept that our flesh is thoroughly corrupted by sin is described theologically as total depravity.
Therefore, our natural situation is somewhat like having a virus living in our entire body, but which especially attacks certain parts (such as a sinus cold or the intestinal flu). The symptoms manifest themselves most strongly in places of special weakness. In a similar way sin, which affects every part of us, may manifest itself in a particular besetting sin, such as gossip, lust, theft, indifference to people’s needs, refusal to believe what God has said, etc. The manifestations of sin will vary from person to person in terms of where it is most evident and how intense it is (Romans 7:5).
The New Nature: Child of God. When a person puts faith in Christ, they are born of the spirit (John 3:4-6). At the moment of regeneration, something comes into existence which had not been there before. That is, a component of the human makeup (the spirit), which was previously dead, is now brought to life. This spirit is born of God and, therefore, is not affected by sin. This is the part of our makeup that loves the Lord, responds to his word and wants to please him.
Sin and the Believer: a Constant Struggle. So why may believers still sin? The answer is simple: because the flesh is not yet dead. In fact, it has not changed in the slightest. It is entirely possible, if they choose, for believers to commit sins that would be typical of the most unrestrained unbelievers, because the dynamic of the flesh is unchanged. That is, the tendencies of the flesh still pull in the same directions they always did (Romans 7:14-18).
That is why the scriptures admonish us to mortify (or kill) the flesh by walking in the spirit (Galatians 5:16, Romans 8:1-4). In other words, moment by moment we choose which nature we will live under. Unbelievers have no choice but to live in the flesh because it is the only nature they have to work with. At times it operates out of the remnants of God’s image and good training, in which case the person may do kind things, good things, noble things, etc. At other times, the flesh may be in a more actively rebellious mode, in which case it may do desperately wicked things. Either way, it can never truly please God because it has no desire or ability to do so (Romans 8:7).
Believers, however, have a choice. Since we now have a new, godly nature within us, we do not have to live in the flesh. As we choose to live under the new nature (walk in the power and wisdom of God’s Spirit) a pattern of godly living is formed in us that grows deeper day by day. The Christian life is one that necessarily involves struggle, and yet it is also one of eventual victory because it depends on God, not primarily on us.
Grace: the Key. One final thing to keep in mind is that the key idea in the gospel is grace. While we grow in faith and godliness, God understands our situation and is patient with our failings. Of course, some people could take this as an excuse for disobedience and immaturity, and so we must guard against this. But it is also reassuring because all of us fail and feel like losers at times. When we do, we simply confess sin, deal with it, and begin to walk in the spirit again in the faith that, as we persevere, God will transform us into the image of Christ.