Cornelius was a Roman Centurion, a commander in the Roman army. Because of this he was a well off citizen of Rome with higher privilege then those around him. Unlike most of his counterparts Cornelius followed Jewish teaching of God and was generous to the poor around him. Because of his faith God uses him as the starting point for the entry of the Gentiles in the Church.
While Cornelius is having a vision Peter is as well. Peter sees a sheet of animals lowered from the sky and is commanded to kill and eat. These animals are all classified as unclean to eat (Leviticus 11). Peter rejects this command to eat the unclean but is told not to call unclean what God has made clean. This wasn't just about being able to eat new foods (like pork!) but extended to people. God was preparing Peter for the coming of Cornelius' servants.
The Jews saw themselves as the chosen people of God, which they were, but they used that to elevate themselves above those around them. Gentiles, a term to mean anyone who was not a Jew, was extended to mean anyone who was a sinner. Just by simply not being Jewish you were considered a sinner. They missed one part of their calling. Yes, they were the chosen people of God but they were chosen to be a "blessing to all nations" (Genesis 12:2-3). There were even rules for how other people groups were to be adopted into the Jewish nation if they accepted God (19:34). God was saying that all who believe in Jesus are the chosen people of God, Jewish or Gentile.
When Peter visits Cornelius' household and hears about Cornelius' vision he puts together what God was telling him in his own vision. He preaches to the Gentiles the good news of Christ, something for which he would be scrutinized by believers later, and the Gentiles accept Christ and receive the Holy Spirit, further solidifying their welcome into the Church.
The church has now shown that it is vastly different from the Jewish faith before in that it now welcomes all people to God. No longer are people classified as clean or unclean when they come to Jesus as God has purified all. God isn't limited to one people but desires all to come and know Him.