What does Hebrews 8:11 mean?

The writer of Hebrews is using a quotation from Jeremiah 31:31–34 to make an important point. God, even while the Old Testament Law was in effect, was already promising to give His people a "new covenant." The nature of this promise is explained in Jeremiah's words, with a special emphasis on the new agreement being something "inside" of the people, rather than something "outside." This is the ultimate distinction between the old and new covenants. Levitical priesthood requires a reliance on laws, obedience, rituals, and ongoing, imperfect sacrifices for sins. Jesus Christ offers a personal, internal, eternal, and completed forgiveness for sins.

As part of this, those who are "brothers" in Christ will "know the Lord." This also speaks to the indwelling of the Holy Spirit, as a guide and teacher for those who are saved (John 16:7–11). The focus of the old covenant was on adherence to rules and laws, taught and interpreted by others. The new covenant focuses on a personal relationship and response to God, through Christ.