1 Samuel 15:22 states, "And Samuel said, Hath the LORD as great delight in burnt offerings and sacrifices, as in obeying the voice of the LORD? Behold, to obey is better than sacrifice, and to hearken than the fat of rams."
Bible sacrifices can be placed into several categories: Old Testament animal sacrifices, the sacrifice of Jesus Christ on the cross, and the spiritual sacrifices made by New Testament believers. There is no need to speak of the sacrifice of Christ in this answer, but we need to look at the other two kinds of sacrifices separately.
Old Testament sacrifices were given as a way for sinful man to approach unto God. This goes all the way back to Abel (Genesis 4:4) or even to the animals killed to provide a covering for the fallen Adam and Eve (Genesis 3:21). When God gave His law to the Jewish people He had chosen as His own, the sacrificial system allowed them to cleanse themselves from their filthiness so that God could continue to dwell among them (Leviticus 1:1-4; 4:35; 16:30; Hebrews 9:13-14). However, the making of sacrifices could not save man because the blood of animals could not put away sin. This is both an Old Testament teaching (Psalm 51:16; Isaiah 1:11; Hosea 6:6) and a New Testament teaching (Hebrews 9:8-10, 10:1-4, 11).
The Old Testament sacrifices provided a way for the Old Testament saints to maintain a right relationship with God. They never gave eternal life. However, that does mean that they were to be taken lightly. Cain discovered to his own destruction that he could not do as he pleased when he brought an offering to the Lord (Genesis 4:3-7). Even under the law, when the offerings were required, what God really wanted was a broken spirit and an obedient heart. Consider these passages:
Psalm 51:16-17 For thou desirest not sacrifice; else would I give it: thou delightest not in burnt offering. The sacrifices of God are a broken spirit: a broken and a contrite heart, O God, thou wilt not despise.
Proverbs 21:3 To do justice and judgment is more acceptable to the LORD than sacrifice.
Isaiah 1:11-16 To what purpose is the multitude of your sacrifices unto me? saith the LORD: I am full of the burnt offerings of rams, and the fat of fed beasts; and I delight not in the blood of bullocks, or of lambs, or of he goats. When ye come to appear before me, who hath required this at your hand, to tread my courts? Bring no more vain oblations; incense is an abomination unto me; the new moons and sabbaths, the calling of assemblies, I cannot away with; it is iniquity, even the solemn meeting. Your new moons and your appointed feasts my soul hateth: they are a trouble unto me; I am weary to bear them. And when ye spread forth your hands, I will hide mine eyes from you: yea, when ye make many prayers, I will not hear: your hands are full of blood. Wash you, make you clean; put away the evil of your doings from before mine eyes; cease to do evil;
Jeremiah 7:22-23 For I spake not unto your fathers, nor commanded them in the day that I brought them out of the land of Egypt, concerning burnt offerings or sacrifices: But this thing commanded I them, saying, Obey my voice, and I will be your God, and ye shall be my people: and walk ye in all the ways that I have commanded you, that it may be well unto you.
Hosea 6:6 For I desired mercy, and not sacrifice; and the knowledge of God more than burnt offerings.
Included with the verse you asked about, 1 Samuel 15:22, these verses (and others that could be added) speak very clearly. God's people were to bring the sacrifices because God commanded it, but God was looking beyond their ceremonial actions into their very hearts.
So, as we come to New Testament application, what are the sacrifices with which God is well pleased? Truly, we are "to offer up spiritual sacrifices, acceptable to God by Jesus Christ" (1 Peter 2:5). Here are a few of those spiritual sacrifices:
- The sacrifice of a broken heart (Psalm 51:17).
- The sacrifice of obedience to the words of God (1 Samuel 15:22).
- The sacrifices of righteous living (Proverbs 21:3; Hosea 6:6).
- The sacrifice of praise and thanksgiving (Hebrews 13:15 - "By him therefore let us offer the sacrifice of praise to God continually, that is, the fruit of our lips giving thanks to his name.")
- The sacrifice of giving (Hebrews 13:16 - "But to do good and to communicate forget not: for with such sacrifices God is well pleased.") Note: to communicate means to make common to others and it includes financial giving. See also Philippians 4:18.
- The sacrifice of ourselves to God (Romans 12:1 - "I beseech you therefore, brethren, by the mercies of God, that ye present your bodies a living sacrifice, holy, acceptable unto God, which is your reasonable service.").
- The sacrifice of a life given in service to others (Philippians 2:17 - "Yea, and if I be offered upon the sacrifice and service of your faith, I joy, and rejoice with you all.").
I am sure that there are other spiritual sacrifices, but this is a good place to start.