Acts 10:18-36,44 reads:

 18 And called, and asked whether Simon, which was surnamed Peter, were lodged there.

 19 ¶While Peter thought on the vision, the ^aSpirit said unto him, Behold, three men seek thee.

 20 Arise therefore, and get thee down, and ^ago with them, doubting nothing: for I have sent them.

 21 Then Peter went down to the men which were sent unto him from Cornelius; and said, Behold, I am he whom ye seek: what is the cause wherefore ye are come?

 22 And they said, Cornelius the centurion, a just man, and one that feareth God, and of good report among all the nation of the Jews, was ^awarned from God by an holy angel to send for thee into his house, and to hear words of thee.

 23 Then called he them in, and lodged them. And on the morrow Peter went away with them, and certain brethren from Joppa accompanied him.

 24 And the morrow after they entered into Cæsarea. And Cornelius waited for them, and had called together his kinsmen and near friends.

 25 And as Peter was coming in, Cornelius met him, and fell down at his feet, and worshipped him.

 26 But Peter took him up, saying, Stand up; I myself also am a^aman.

 27 And as he talked with him, he went in, and found many that were come together.

 28 And he said unto them, Ye know how that it is an unlawful thing for a man that is a Jew to keep company, or come unto one of another nation; but God hath shewed me that I should not call any man ^acommon or ^bunclean.

 29 Therefore came I unto you without gainsaying, as soon as I was sent for: I ask therefore for what intent ye have sent for me?

 30 And Cornelius said, Four days ago I was ^afasting until this hour; and at the ninth hour I prayed in my house, and, behold, a^bman stood before me in bright clothing,

 31 And said, Cornelius, thy prayer is heard, and thine alms are had in remembrance in the sight of God.

 32 Send therefore to Joppa, and call hither Simon, whose surname is Peter; he is lodged in the house of one Simon a tanner by the sea side: who, when he cometh, shall speak unto thee.

 33 Immediately therefore I sent to thee; and thou hast well done that thou art come. Now therefore are we all here present before God, to hear all things that are commanded thee of God.

 34 ¶Then Peter opened his mouth, and said, Of a truth I perceive that God is no ^arespecter of persons:

 35 But in every ^anation he that ^bfeareth him, and ^cworketh^drighteousness, is ^eaccepted with him.

 36 The ^aword which God sent unto the children of Israel, preaching peace by Jesus Christ: (he is Lord of all:)

 44 ¶While Peter yet spake these words, the Holy Ghost fell on all them which heard the word.

Questions about the reading:

When Peter clearly understood the meaning of the vision, what did he declare? 

He declared that God is no respecter of persons, but that every nation that fears him, and works righteousness, is accepted with him.

What does it mean that God is “no respecter of persons?” 

I believe that it means that God does not have a care for what a person is. If they be rich or poor, black or white, bond or free, they are all his children and he loves them.

How did the Holy Ghost assist the Jewish Saints in understanding that the blessings of the gospel were to be available to all people at that time? 

He was praying and fasting to the Lord and an angel of the Lord came down and told him to seek out Peter, and Peter was able to room him and tell him of the things the Lord had wanted them to learn.