The overall theme of Paul's Letter to the Galatians is covered extensively in the Letter to the Romans – but I must confess that I find it in some ways easier to follow here. Paul is intensely angry at those he perceives as offering "false" and "perverted" teachings to the congregations he himself founded, and that anger seems to sharpen his thinking as well as his tongue.
At issue is whether followers of Jesus must be Jews – i.e., must non-Jewish converts, Gentiles who are being baptized, be circumcised as well? Circumcision is the mark of identity as a Jew, and some teachers appear to have come into the congregations in Galatia (which seem to be four churches in central Asia Minor, though there is some dispute about this) arguing that Gentiles must first be circumcised as well as baptized. Two perspectives on this issue are important, ours and Paul's. From our perspective, we can see that circumcising Gentile converts would mean that "Christianity" was to remain a movement within Judaism; Paul and others are arguing that belief in Christ constitutes a new relationship with the same God, and therefore new beliefs and new practices. This amounts to a new religion, though Paul himself does not ever put it in quite those terms. This dispute, then, is an extremely important historical insight into the birth of the Christian Church.
Paul's perspective on this issue is both profoundly theological and deeply pastoral. He argues that Abraham believed in God's promise; we are children of Abraham, then, through believing that same promise of God to be fulfilled in Christ. The law – the Ten Commandments, and the other 603 found in the Old Testament – was given when the people of God needed behavioral guides, but now the law serves to point up our shortcomings, our inability to keep all of the commandments all of the time. Under the covenant of the law, then, we are subject to rejection by God, and thus (in Paul's view) death. The law cannot save us. Instead, we are freed from the law by the death of Christ, by God taking on the shortcomings – sinfulness – of human nature himself in Jesus, and thus we are promised in his resurrection the same eternal life he himself enjoys. Our freedom in Christ, Paul believes, is so great, and is so important to making our life in the community of the faithful even possible, that we must not allow ourselves to fall back into an economic (you-do-this-and-I'll-do-that) relationship with God that we cannot possibly maintain. Instead, we respond to the freedom given in Christ by attempting to lead lives based on love, knowing that even when we fail we are welcomed back into the love of the community because of the love of God.
My explanation may be muddling things even more – perhaps I'm not angry enough to be as sharp as Paul! Galatians repays a more than cursory reading – I recommend the commentaries of the Oxford Annotated Bible or Oxford's Access Bible as easy guides to Paul's argument.