"Now while Peter wondered within himself what this vision which he had seen meant, behold, the men who had been sent from Cornelius had made inquiry for Simon's house, and stood before the gate. And they called and asked whether Simon, whose surname was Peter, was lodging there. While Peter thought about the vision, the Spirit said to him, 'Behold, three men are seeking you. Arise therefore, go down and go with them, doubting nothing; for I have sent them.' Then Peter went down to the men who had been sent to him from Cornelius, and said, 'Yes, I am he whom you seek. For what reason have you come?' And they said, 'Cornelius the centurion, a just man, one who fears God and has a good reputation among all the nation of the Jews, was divinely instructed by a holy angel to summon you to his house, and to hear words from you.' Then he invited them in and lodged them. On the next day Peter went away with them, and some brethren from Joppa accompanied him" (Acts 10:17-23).
As Peter wondered about the meaning of the vision, Cornelius' servants arrive at Simon's house and call for Peter. Peter was then instructed by the Holy Spirit to go with these Gentile men, without wavering, for they were sent to Peter by God (indirectly through Cornelius). Peter speaks with the men and asks them about the reason for their arrival. They described their master, Cornelius, to Peter as a "centurion, a just man, one who fears God and has a good reputation among all the nation of the Jews." They explain that an angel had instructed Cornelius to summon Peter to his house so that he could preach to them. Surely Peter is starting to understand the significance of the vision. Had he not had the vision and explicit instructions from the Spirit, it is almost certain that Peter would not have invited these men in (as he ended up doing) and he certainly would not have gone with them the next day (as he also did). Being hospitable toward Gentiles and traveling with them is another step toward Peter understanding God's will. Some other Christians also traveled with the group from Joppa to Caesarea.
"And the following day they entered Caesarea. Now Cornelius was waiting for them, and had called together his relatives and close friends. As Peter was coming in, Cornelius met him and fell down at his feet and worshiped him. But Peter lifted him up, saying, 'Stand up; I myself am also a man.' And as he talked with him, he went in and found many who had come together" (Acts 10:24-27).
Cornelius, eagerly anticipating an important meeting with Peter and knowing about how long such a journey would take, had assembled his family members and close friends together in his home (a large group of people in all). He wants them all to hear whatever he is about to hear from Peter. Such excitement and passion for evangelism is refreshing (and it came from a non-Christian no less!). Upon meeting Peter, Cornelius prostrated himself and attempted to worship the apostle. Peter would have none of that, however. He lifted Cornelius back to his feet and told him that he was only a man.
As a side note, our Catholic friends should be asked at this juncture: "Why, if Peter is the first 'pope' as is claimed by Catholics, did he refuse to accept Cornelius' worship when the popes for centuries have accepted such adoration, claiming to be God's representatives on Earth?" Could it be that Peter was not a pope? Could it be that no mortal today should accept worship (no matter what title he is given or takes upon himself)? I believe the answer to both of these questions is a resounding "yes" (cf. Rev. 22:8,9).