The Savior instructs through parables. In chapter 15 of Luke, there are three instances.
What is the significance?
This subject has captured the imagination of many classical and contemporary artists. The paintings are dramatic and poignant.
Some of the talks from Church General Authorities also reference these stories. Elder Thomas S. Monson referred to these parables as "Search and Rescue". His talk was about humanitarian efforts.
The second story tells about a woman who misplaces one of her ten coins, turns the house upside down to find where it was lost, and rejoices when she finds it.
There is rejoicing in each instance when that which was lost is recovered.
The Prodigal Son story differs from the others. Jesus makes it quite clear that the son in the parable parted company from his father and his father's household thinking he would be better off without them. He takes some division of fortunes from his inheritance and goes out into the world.
His money is all squandered in "riotous living". That phrase has interesting significance in today's world.
Interesting also that the story differs in that the father does not leave his establishment to go off seeking after his lost son. In this story the father and son apparently parted company by mutual agreement.
Instead of investing his wealth wisely, the son eventually falls on hard times, and returns to his father's house hoping to receive treatment at least equal to the standard of living afforded to the father's servants.
I am curious, why is this story different from the parable of lost sheep?
Perhaps we have a different obligation toward those who innocently wander astray than we do for those who wilfully leave the company of the disciples of Jesus?
In both instances the lost are welcome to return, but in the case of the sheep, the shepherd leaves the ninety and nine, and goes actively searching to rescue the lost lamb, bringing it back to the fold carrying the lost sheep on his shoulders.
This contrasts with the prodigal son, who eventually comes to regret the misfortune he experiences, and returns to his father's house voluntarily. He has lost all hope, and will confess to his father the heartrending realization, "I am no more worthy to be called thy son".
The father, seeing his returning son approach from afar, rejoices in his return and kills the fatted calf in celebration.
One sour note celebrating the return of the prodigal son. The older brother complains, jealous and angry to see all the attention is being given to his younger brother, when he was the one who remained ever faithful.
The father admonishes him, "Son, thou art ever with me, and all that I have is thine.It was meet that we should make merry, and be glad: for this thy brother was dead, and is alive again; and was lost, and is found."
Interesting choice of phrases in this reproach. "Was dead, and is alive again, was lost, and is found". What possible significance is there in these words?
Jesus apparently left the parable rest with that point. We do not learn if the errant prodigal is restored to a share of the inheritance, or if the father later installs him as one of the household servants. Perhaps it is sufficient to know assurance that the elder brother's reward for his uninterupted faithfulness will not be compromised by the return of the prodigal.