When you look at the word sacrifice in the Old Testament, the Hebrew word that is used means an animal killed for an offering. Once instance where both the words sacrifice and offering are used are found in Psalm 40:6 (NKJV)
6 Sacrifice and offering You did not desire; My ears You have opened. Burnt offering and sin offering You did not require.
The sacrifices required by God...the death and shedding of blood ended when Christ became the ultimate lamb sacrificed for our sins. However, we do find in this: Romans 12:1 (NKJV)
1 I beseech you therefore, brethren, by the mercies of God, that you present your bodies a living sacrifice, holy, acceptable to God, which is your reasonable service.
We no longer need to sacrifice animals etc. but we present ourselves as LIVING sacrifices. This means as a living sacrifice some things are to die in our lives: For instance: we let our hands work no sin. It dies. We allow our eyes to look upon no evil. We let some things we may have watched at one time, die...not causing our eyes which affect our mind and actions to sin. It is a dying to self which Jesus spoke of to be His disciple.
An offering according to the Hebrew word used in Psalm 40:6 speaks of the act of making a gift or a free contribution; what is brought to another as a gift. This word is also used in Malachi 3:8-10 when the Lord spoke of the giving of our tithes and offerings.
Hope this helps. Blessings!