1 Samuel 15:22And 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.
See All... - What kind of sacrifice pleases God? What does He reject?

1 Samuel 15:22And 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.
See All... 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 4:4And Abel, he also brought of the firstlings of his flock and of the fat thereof. And the LORD had respect unto Abel and to his offering:
See All...) or even to the animals killed to provide a covering for the fallen Adam and Eve (Genesis 3:21Unto Adam also and to his wife did the LORD God make coats of skins, and clothed them.
See All...). 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 [1] And the LORD called unto Moses, and spake unto him out of the tabernacle of the congregation, saying, [2] Speak unto the children of Israel, and say unto them, If any man of you bring an offering unto the LORD, ye shall bring your offering of the cattle, even of the herd, and of the flock. [3] If his offering be a burnt sacrifice of the herd, let him offer a male without blemish: he shall offer it of his own voluntary will at the door of the tabernacle of the congregation before the LORD. [4] And he shall put his hand upon the head of the burnt offering; and it shall be accepted for him to make atonement for him.
See All...; 4:35
See All...; 16:30
See All...; Hebrews 9:13-14 [13] For if the blood of bulls and of goats, and the ashes of an heifer sprinkling the unclean, sanctifieth to the purifying of the flesh: [14] How much more shall the blood of Christ, who through the eternal Spirit offered himself without spot to God, purge your conscience from dead works to serve the living God?
See All...). 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 51:16For thou desirest not sacrifice; else would I give it: thou delightest not in burnt offering.
See All...; Isaiah 1:11To 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.
See All...; Hosea 6:6For I desired mercy, and not sacrifice; and the knowledge of God more than burnt offerings.
See All...) and a New Testament teaching (Hebrews 9:8-10 [8] The Holy Ghost this signifying, that the way into the holiest of all was not yet made manifest, while as the first tabernacle was yet standing: [9] Which was a figure for the time then present, in which were offered both gifts and sacrifices, that could not make him that did the service perfect, as pertaining to the conscience; [10] Which stood only in meats and drinks, and divers washings, and carnal ordinances, imposed on them until the time of reformation.
See All..., 10:1-4
See All..., 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 [3] And in process of time it came to pass, that Cain brought of the fruit of the ground an offering unto the LORD. [4] And Abel, he also brought of the firstlings of his flock and of the fat thereof. And the LORD had respect unto Abel and to his offering: [5] But unto Cain and to his offering he had not respect. And Cain was very wroth, and his countenance fell. [6] And the LORD said unto Cain, Why art thou wroth? and why is thy countenance fallen? [7] If thou doest well, shalt thou not be accepted? and if thou doest not well, sin lieth at the door. And unto thee shall be his desire, and thou shalt rule over him.
See All...). 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 [16] For thou desirest not sacrifice; else would I give it: thou delightest not in burnt offering. [17] The sacrifices of God are a broken spirit: a broken and a contrite heart, O God, thou wilt not despise.
See All... 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 21:3To do justice and judgment is more acceptable to the LORD than sacrifice.
See All... To do justice and judgment is more acceptable to the LORD than sacrifice.

Isaiah 1:11-16 [11] 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. [12] When ye come to appear before me, who hath required this at your hand, to tread my courts? [13] 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. [14] Your new moons and your appointed feasts my soul hateth: they are a trouble unto me; I am weary to bear them. [15] 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. [16] Wash you, make you clean; put away the evil of your doings from before mine eyes; cease to do evil;
See All... 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 [22] 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: [23] 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.
See All... 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 6:6For I desired mercy, and not sacrifice; and the knowledge of God more than burnt offerings.
See All... 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 15:22And 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.
See All..., 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 2:5Ye also, as lively stones, are built up a spiritual house, an holy priesthood, to offer up spiritual sacrifices, acceptable to God by Jesus Christ.
See All...). Here are a few of those spiritual sacrifices:

1. The sacrifice of a broken heart (Psalm 51:17The sacrifices of God are a broken spirit: a broken and a contrite heart, O God, thou wilt not despise.
   See All...).
2. The sacrifice of obedience to the words of God (1 Samuel 15:22And 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.
   See All...).
3. The sacrifices of righteous living (Proverbs 21:3To do justice and judgment is more acceptable to the LORD than sacrifice.
   See All...; Hosea 6:6For I desired mercy, and not sacrifice; and the knowledge of God more than burnt offerings.
   See All...).
4. The sacrifice of praise and thanksgiving (Hebrews 13:15By him therefore let us offer the sacrifice of praise to God continually, that is, the fruit of our lips giving thanks to his name.
   See All... - "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.")
5. The sacrifice of giving (Hebrews 13:16But to do good and to communicate forget not: for with such sacrifices God is well pleased.
   See All... - "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
   See All....
6. The sacrifice of ourselves to God (Romans 12:1I 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.
   See All... - "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.").
7. The sacrifice of a life given in service to others (Philippians 2:17For the violence of Lebanon shall cover thee, and the spoil of beasts, which made them afraid, because of men's blood, and for the violence of the land, of the city, and of all that dwell therein.
   See All... - "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.