Praised be Jesus and Mary, now and forever.
Yesterday we heard part of Jesus' Good Shepherd discourse. Today's Gospel is the beginning of that discourse, and we'll hear another part of it tomorrow. Christ begins with the parable that no one understands at first. In the parable, the sheepfold is the Church, which belongs to God.
Honest people enter a sheepfold through the gate, instead of climbing over some other way, and the gateway to the Church is Christ, or faith in Him. Those who do not come through Christ are thieves and robbers. They come to steal God's sheep for themselves, or for those they serve. Thus the devil and his servants work to spread false religion and heresies.
The door or gate, which closes the gateway is the Holy Scripture, which in the time of Christ was called the Law and the Prophets. A gate can both open to let someone in, and be closed to keep someone out. Thus Scripture on the one hand opens the way to faith in Christ, by which we enter the Church, and on the other hand, protects the Church by excluding moral and doctrinal errors. The doorkeeper who opens the Scriptures is the Holy Spirit, for faith is His gift. Christ calls His sheep by name. He knows us better than we know ourselves. He calls us to believe in Him, to recognize Him as worthy of faith, and to follow Him wherever He goes. The sheep effectively say, this is my shepherd, who will lead me to the pastors of eternal life, and they follow Him.
So the sheep are all the faithful in the Church, all those who follow Christ. The true pastors are those who enter through Christ. The gatekeeper, who is the Holy Spirit, opens to them and gives them authority, so that what they do is ratified by God. They lead the sheep, that is, the faithful to the pastors of sound doctrine, grace, and virtue. They go ahead of them, or at least they should, by giving a good example of Christian life. They call the sheep by name, so that they may care for them individually, exhort them, stimulate them, and even compel them to live in accord with the teaching of Christ. Today's Gospel ends with Christ saying, I came that they might have life and have it more abundantly. We have the true life, which is eternal life, by baptism, and in the life of grace. We will have it more abundantly in heaven. In yesterday's Gospel, Christ said, I have other sheep that do not belong to this fold. When He said they do not belong to this fold, He meant that they were not Israelites. Cornelius was one of those other sheep. Christ said, these also I must lead, and they will hear my voice, and there will be one flock, one shepherd. The one flock is the Catholic Church, and the one shepherd is Christ, whose vicar is the Pope. Through the Church, we hear the voice of Christ, and we follow this voice.
To this vicar, Saint Peter, a vision was given to bring about what Christ had foretold.
He was still at Joppa, where he had raised Tabitha from the dead. The previous day, an angel had been sent from God to a sanctuary named Cornelius. He was uncircumcised, but he feared God. The angel revealed to him that God had taken into account his prayers, had taken note of his prayers and alms, and so he should send to Joppa, have Peter brought to him, and listen to what he had to say. He didn't know Peter, so the angel gave instructions so that the messengers could find him. As they were arriving, Saint Peter had the vision we heard described. What this is about is the entry of the Gentiles into the Church, without the requirement to be circumcised and to observe the rest of the mosaic law. Jesus had already taught this, we find it in the Gospel written by Saint Peter's translator. In Mark 7:18-19, Jesus said, Do you not see that whatever goes into a person from outside cannot defile him, since it enters not into his heart, but his stomach, and is expelled? Thus he declared, All foods clean, the evangelist adds. But this was a bit too subtle, so Peter was given a clearer sign.
He saw animals of all kinds lowered in a sheet and was told to kill and eat. These were not clean animals allowed as food by the mosaic law, so he refused. But the voice from heaven replied, What God has made clean, you are not to call profane. While it is true that God has now made all foods clean, so that it is no longer necessary for Christians to eat only kosher food, the animals are primarily symbolized people. Vices can make human beings like pigs or serpents, foxes or various birds. This is the same image used in Isaiah chapter 11, The wolf shall dwell with the lamb, and the leopard shall lie down with the young goat.
Those who formerly preyed on other men will live in peace with them in the Church, once they become docile disciples of Christ. As the Gentiles extend to the four corners of the earth, so the animals are in a sheet let down by four corners. Saint Peter is told to kill and eat. What we eat becomes part of us, so the eating signifies incorporation into the Church, Saint Peter explains. Before this can happen, he must kill them, that is, he must make them die to sin by preaching the word. The animals are lowered in the sheet three times, because they must go under the baptismal water three times, as they are baptized in the name of the Father, and of the Son, and of the Holy Spirit. Saint Peter didn't understand the meaning of the vision at first, but when Cornelius' messengers arrived, the Holy Spirit told them to go with him.
With six companions, he followed them to Cornelius' house in Caesarea. There he entered the house of the uncircumcised man, even though this was forbidden to Jews, because God had taught him not to call any man unclean. He preached the Gospel to Cornelius, his relatives and his close friends who had been called there for the occasion. As he was speaking to them, the Holy Spirit came upon them. This was a clear sign that they could receive baptism without any further preparation.
So he baptized them without circumcising them or imposing on them the obligation to observe the law of Moses. Today's reading talked about the resulting controversy and how it was resolved by simply explaining what happened. It was evident that this was God's idea, not Peter's, and so those who had objected at first then glorified God for his amazing mercy. 
Lastly, I share an interpretation of Jesus' parable that's a little different from the standard one that I gave you at the beginning of the homily. I take it as an unconfirmed but interesting theory.
There are some things about the parable that make it hard for any one interpretation to fit.
For example, Jesus is first the shepherd that enters through the gate, and then he is the gate.
Some of the odd thing can be explained by the fact that Christ may be adjusting the parable because it wasn't understood at first, or because he's expressing different aspects of himself.
But one odd thing is this. If the sheepfold is the church, why does the shepherd call the sheep by name and lead them out? Does a good shepherd lead us out of the church?
St. Augustine's answer is that he leads us out when we go to heaven.
But I would suggest that the sheepfold is instead the Mosaic Law, within which the Jewish people were enclosed. In fact, Christ says, I have other sheep that do not belong to this fold, and everyone recognizes that these others are the Gentiles. The Mosaic Law is the enclosure that kept the Jews safe during the long night before the coming of Christ, the Son of Justice.
Then he called his sheep by name to leave the enclosure and follow him to pasture.
He took away the burden of the old law and repressed them with the sacraments and the full revelation. Through his church, he then called other sheep who were not in the enclosure.
If the enclosure is the Mosaic Law, then to enter through the gate is to fulfill the law and the prophets. The false Christs who came before and after him did not do this. Or again, the gate could be his virgin mother, who is called a gate in Ezekiel 44. The Prince of Peace alone passed through this gate, and it remains forever closed, for she is forever a virgin.
On this interpretation, the false Christs did not come through the gate because none of them was born of a virgin. However, we may interpret the parable, it is clear that we must listen to the voice of the Good Shepherd and follow him wherever he leads us. Let us thank him for his great mercy and profit by following him faithfully and docilely. 
Praised be Jesus and Mary, now and forever.