The Old Covenant, unlike the new, was a physical (earth) based agreement that was initially made at Mount Sinai with the children of Israel. It required obedience to the letter of God's law.
The Old Covenant promised that a person or a nation would be physically blessed (health, wealth, etc.) or cursed (e.g. poverty, captivity, etc.) depending on their obedience to the Eternal's commandments and judgments (Exodus 19:5 - 6, Deuteronomy 28, Leviticus 26). It was the Being who became Jesus Who established the Old Covenant with Israel.
Unlike the Old Covenant, the new one (first fully defined in Matthew 5 - 7) is spiritually based. It is founded on obedience from the heart and a fulfilling of God's laws not only in the letter but also in their spiritual intent.
The New Covenant promises spiritual blessings (John 3:16, Hebrews 8:6, 9:15, Romans 8, Luke 18:29 - 30), such as eternal life full of love and glory for those who are obedient. Additionally, unlike the Old Covenant, the new one promises eternal death to those who knowingly reject submitting to the Father.
The Old Covenant, unlike the New (see our article on the Jerusalem Conference) required physical circumcision (Genesis 17:9, Exodus 12:48, John 7:22) and people under it were not (with a few exceptions) given God's Spirit (Romans 7:6, Hebrews 8:7 - 9, Matthew 5:21 - 45).
The New Covenant, in contrast to the Old, requires circumcision (repentance) of the heart (Acts 2:38, 3:19, Romans 2:25 - 29, Philippians 3:3, Colossians 2:11 - 13) and the Holy Spirit is made freely available to believers (Matthew 5:21 - 48, Romans 7:6, Galatians 2:20, Hebrews 8:10 - 12).
Under the Old Covenant system, physical works of law such as animal sacrifices were required (Leviticus 1 - 7). These sacrifices were made to justify a person but only to an earthly tabernacle (Hebrews 5:1 - 3, 9:1 - 13). A human High Priest served in the tabernacle as an intercessor between God and man (Hebrews 8:3 - 5, Exodus 28:1).
Under the New Covenant, Jesus' sacrifice, the fruits of God's spirit and good works are required (Hebrews 9:10, 10:12, 13:15, Galatians 5:22 - 24, Ephesians 2:10, Romans 12:1 - 2, Matthew 19:17, Mark 15:15 - 18).
Christ's sacrifice, in God's heavenly temple, forgives and justifies those who repent and are baptized (Hebrews 7:25 - 27, 9:23 - 28 Acts 11:18, Galatians 2:20, Romans 3:24 - 25). Jesus continually serves as man's spiritual High Priest and intercessor (Hebrews 4:14 - 16, 5:1 - 9, 7:17, Hebrews 8:1 - 2, 9:11).
Under the Old Covenant, if the letter of the law was obeyed, a person could access God through an earth-based tabernacle (Hebrews 9:13, Galatians 3:12, Leviticus 18:5, 2 Corinthians 3:16). No direct access to him was available or offered (Hebrews 9:6 - 8, 10:19).
The New Covenant offered by Christ, however, unlike the Old Covenant, gives those who obey one-on-one access to the Father through him (John 4:23 - 24, 16:23 - 27, Hebrews 6:20, 10:23, Romans 8:9 - 39, Ephesians 2:18 - 22, 2 Corinthians 3:6). This means that believers can talk to him at any time no matter where they are (Matthew 27:51, Hebrews 4:14 - 16, Ephesians 7:25 - 27, John 16, Galatians 4:6 - 7, Romans 8:15, Ephesians 2:18).
Lastly, those who were in the Old Covenant agreement could not have their consciences made clean or perfect before God in heaven (Hebrews 9:9, 10:4, 11). In the New Covenant made possible through Jesus' blood, however, a believer's sins can be spiritually forgiven. The Holy Spirit within them is able to purge their consciences and make them wholly acceptable to the Father (Hebrews 10:14 - 17, 2Corinthians 3:9, Colossians 1:27, Matthew 5:48, etc.).