The girl was joyful, beautiful, she shown bright as gold, but she was careless. In her tangled mess she played with the ball and one day, carelessly threw it through a window.
When she realized what she had done, her response was immediate. She ran to the owners and threw herself at their feet. She had such remorse, such purity of grief that she was immediately forgiven. She still has to pay for the window though.
Offerings can be made to offset debt or correct wrongs. Not necessarily even speaking of evil acts or karmic balance as some may think here, but those incorporating poor choices.
Sacrifice and offerings can also be done for others. This should be done with humility and selflessness and most importantly, quietly.
The man had little, but he had more than some others and so used what he had to help those beneath him. A few noticed and when they saw he needed help to continue his work, they gathered together. They collected items he needed and left them on his stoop in the dark of the night.
He found the items the next morning and rejoiced. He didn’t know who left them so the sun shone brightly over the whole world.
Do note that selflessness is vital here. The person may not need the blessings you think they need, their course may change in ways you don’t anticipate. The concept of Quiet is important in many cases. The prayers and offerings are not for you, they are not to gain acclaim or credit for having done so- this can remove the selflessness of the act. Of course there will be times quietness isn't necessary such as prayer for a person who requests, etc.
Ultimately, sacrifice and offerings are about change. Change in self, relationships, and the world about you. Make offerings, but know why.