Here Paul likens Isaac's birth to the new-covenant relationship, and Ishmael is compared to the old covenant. How do the covenants relate to these two sons of Abraham? God told Abraham he would have a son by Sarah. Because Sarah was past age, Abraham did not believe it possible for the promise to be fulfilled. So, falling back on the old-covenant principle of trying to do it in human strength and devising, Abraham took a concubine, Hagar, to help things along. The son born of this arrangement was likened to the old-covenant idea of "we will do."
When Isaac was born of Sarah, it was a miracle of grace. God actually brought supernatural life into a dead womb so that Isaac could be born. This represents the regenerating miracle of grace that makes obedience possible under the new covenant. It does not depend on poor promises of man, but on the unfailing assurance of God. "I will put my laws into their mind, and write them in their hearts: and I will be to them a God." Hebrews 8:10-12. Only by the indwelling miracle grace of God can the law be kept.
One fact appears in this allegory of the two covenants. The child of promise represents the new covenant because Abraham obeyed God and followed His instruction in begetting that miracle child. Those under the new covenant are those who obey God's commandments. Ishmael represents disobedience to God's way. Commandment-breakers are the ones who are operating under the old covenant.