In today's reading, Paul takes an old story and allegorizes it. OK, you may be asking yourself, what is an allegory? An allegory is a spiritual or symbolic interpretation of a literal story. Today's literal story is the one about Abraham, Sarah, and Hagar, the latter two being the free woman and the slave woman.
At this point you may be asking yourself, where is this going? That is a very good question. Let me try to tie Paul's allegory back into the message.
Sarah and Hagar illustrate the conflict between law and grace. These two women represent God's two covenants. These covenants are of course the Mosaic Covenant, or the law, which is represented by Hagar, the slave woman. The second covenant is the Grace Covenant, the free gift that God gave us in the death and resurrection of Jesus Christ. This second covenant is represented by Sarah, the free woman.
In verse 26, Paul uses the phrase "she is our mother" to describe Sarah to show that faith, not adherence to the law, is the source of our salvation. While some Jews were claiming that one had to be a descendant of Abraham and adhere to the law to find salvation, Paul was arguing to the Galatians that they could be descendants of Abraham through their faith in Jesus Christ as their one and only savior.
Just as Isaac's birth was a miracle of God, so Christianity, by offering people the opportunity to be born again, is a miracle of God. Just as Isaac's mother, Sarah, was free, Isaac was free, and so Christianity offers true freedom because it depends not on our actions but on God's unchangeable promises to us.