From the time of Adam to the time of Moses, and from Moses to Jesus Christ, the Lord's covenant people were commanded to sacrifice the first fruits of the field and the first-lings of the flock unto the Lord. Upon the death, Atonement, and Resurrection of the Lord, sacrifice by the shedding of blood was ended, for no greater blood could be shed than that of the Lamb of God.
However, while we do not perform burnt or wave offerings, the law of Sacrifice is still in effect. Instead of sacrificing livestock, we sacrifice money, time, talents, and anything with which the Lord has blessed us.
Most of all, we sacrifice a broken heart and a contrite spirit. What are these things? The ancients zealously and exactly adhered to the sacrificial ordinances they were given; how do we sacrifice as zealously and exactly? In other words, what is the nature of a broken heart and a contrite spirit?
First, a broken heart recognizes the need for the Savior and His Atonement. It is a feeling of remorse for the way we've messed up our lives, and a recognition that we need to be somewhere else. A broken heart does not try to hold onto a little bit of sin, but is willing to put it all away, even when it is uncomfortable.
Second, a contrite spirit is the willingness and desire to start accepting the grace that the Lord would give to us, to accept the Lord's sovereignty over our lives and to start acting according to His commandments. A contrite spirit makes and keeps sacred covenants with the Lord without trying to impose one's owns conditions, timelines, or exceptions to such covenants.
Together, a broken heart and a contrite spirit make us give up the things of the world. Perhaps the most difficult sacrifice we are required to make is to sacrifice control over our own lives. No longer are we able to walk our own paths; with a broken heart and a contrite spirit, we are forced to abandon our own strange paths and walk the strait and narrow path defined only by God.
Indeed, the greatest sacrifices, the most difficult, all entail a loss of control: Abraham was willing to sacrifice Isaac, even though he wouldn't be able to control his legacy afterwords; Peter and his brethren sacrificed their fishing despite the financial uncertainty this created for their families, and so on.
The end result of these sacrifices is more and greater joy than we ever have here, but in the temporal blinders of the here and now, the eternal consequences sometimes seem distant and/or opaque. So giving up control of our own lives and defining them based on the will of the Lord is difficult. But it is worthwhile, even necessary.