And he said, A certain man had two sons: And the younger of them said to his father, Father, give me the portion of goods that falleth to me. And he divided unto them his living.
And not many days after the younger son gathered all together, and took his journey into a far country, and there wasted his substance with riotous living.
And when he had spent all, there arose a mighty famine in that land; and he began to be in want. And he went and joined himself to a citizen of that country; and he sent him into his fields to feed swine. And he would fain have filled his belly with the husks that the swine did eat: and no man gave unto him.
And when he came to himself, he said, How many hired servants of my father's have bread enough and to spare, and I perish with hunger! I will arise and go to my father, and will say unto him, Father, I have sinned against heaven, and before thee, And am no more worthy to be called thy son: make me as one of thy hired servants.
And he arose, and came to his father. But when he was yet a great way off, his father saw him, and had compassion, and ran, and fell on his neck, and kissed him.
And the son said unto him, Father, I have sinned against heaven, and in thy sight, and am no more worthy to be called thy son.
But the father said to his servants, Bring forth the best robe, and put it on him; and put a ring on his hand, and shoes on his feet: And bring hither the fatted calf, and kill it; and let us eat, and be merry: For this my son was dead, and is alive again; he was lost, and is found. And they began to be merry.
Jesus Christ, Luke 15:11
To kill the fatted calf means proverbially to pull out all the stops in celebrating something or someone. When you visit your friends and they take you to the nicest restaurant in town and buy a nice bottle of wine, they are in essence killing the fatted calf.
The parable of The Prodigal Son is almost as famous as Jesus himself. In it, the younger foolhardy son takes his inheritance and spends it poorly ("on harlots", woo!). Broke and hungry, he returns home where his father greets him with open arms and literally kills the specially fed calf for a homecoming feast. When the older son chastises his father, pointing out how ungrateful the son has been while he has been faithful, the father looks at his oldest son with sad eyes and says, "You have always been with me, and whatever is mine is yours, too. Now rejoice! For your brother has returned from the dead!"
Jesus' point is that we should never turn our backs on people who are willing to admit they were wrong, but instead should welcome them as strongly as ever. There is of course the second part to the allegory - that God is the father, and Jesus is the fatted calf, and we are all his prodigal sons. Even when we sin, if we return to God, admit we were wrong, and ask for forgiveness, he will welcome us with open arms and a warm heart.
Some people, either having missed the point of the story or conversely having gotten it a little too clearly, use this phrase sarcastically to express anger at people who have wronged them. For example: "What, you think I should kill the fatted calf just because you said you're sorry?" Whatever the connotation may be, the "fatted calf" has a longstanding religious history in the English language.