Biblically, blood is a symbol for life. The best-known application of this is that blood provides atonement, where one life symbolically pays the life-debt of another. However, the various covenants show a second application, where blood represents life given as a pledge of faithfulness. God ratified the covenant with Israel with blood, and those sacrificial animals gave their lives to symbolize life given as a pledge. Significantly, the blood designated as the blood of the covenant at Mt. Sinai did not come from a sin offering, but from burnt and peace offerings (Exodus 24:4-8). That covenant was sealed before the first sin offering was commanded (Exodus 29:14).
The New Covenant is also sealed with blood, but it is not sprinkled on the outside of those making the covenant, as happened with Israel. Instead, it is ingested into the innermost parts of the person. Rather than being sealed with the blood of oxen, the New Covenant is sealed with infinitely more precious blood, blood that serves as a testimony of eternal life (see I John 5:6-13), as well as a pledge of God's loyalty to those within the covenant:
Notice that God makes us complete through the blood of the covenant. Christ's blood is a pledge that God made that He will finish His extraordinary purpose - one that goes beyond forgiveness and culminates in our spiritual completion. When we are complete, then Passover will be fulfilled. But the forgiveness of sins comes through the covenant, not before it. Before the covenant, God overlooks - He passes over. When we pledge our loyalty to God through baptism, and accept His covenant after repenting, He then forgives us. When we are put into Christ, we are washed clean (see Acts 2:38; 4:12; 8:36-37; 22:16; Romans 6:3-7, 23; Colossians 2:12; Titus 3:4-7). But the cleansing blood of atonement is only available to those who accept the divine blood of the covenant.
Forgiveness is part of the covenant because we need God's forgiveness throughout the process of being made complete. Neither the Old nor the New Covenant - nor the covenant with Abraham - were preceded by atonement. Instead, God makes covenants with those whose transgressions He has passed over. It is within the covenant, then, that sin is addressed. This is why Jesus proposed the New Covenant to His disciples at that Passover observance even before He died to provide atonement the following afternoon.
However, the various covenants show a second application of the symbol of blood, where blood represents life given as a pledge of faithfulness. When God made the covenant with Israel, He had them seal it with blood, which we will see in Exodus 24:
The source of the blood used here is significant. The blood designated as the blood of the covenant did not come from a sin offering. In fact, the first sin offering is not commanded until 5 chapters later (Exodus 29:14). Instead, verse 5 here tells us that the blood of the covenant came from burnt offerings and peace offerings. If you recall what the offerings picture, the burnt offering symbolizes wholehearted devotion to God, which certainly fits with the ratification of a covenant. And the peace offering pictures a relationship that is on good terms, such that all the participants are satisfied and can enjoy the abundance of the occasion. It is easy to see why the blood that sealed the Old Covenant was taken from these two offerings.