Acts 10:37-42
Acts 10:37 (KJB)
That word, I say, ye know, which was published throughout all Judaea, and began from Galilee, after the baptism which John preached;
As Peter continues to bring the Gospel message to the household of Cornelius, he reminds them that they already know the facts about the birth, death, and resurrection of the Lord Jesus Christ and that message was continually preached from the time of John the Baptist when he introduced the Lord Jesus Christ after he had been preaching the Gospel of repentance. Now Peter would be able to give a more detailed description of the Gospel message. It is one thing to know the facts about it but now they will learn the more intimate details.
Acts 10:38 (KJB)
How God anointed Jesus of Nazareth with the Holy Ghost and with power: who went about doing good, and healing all that were oppressed of the devil; for God was with him.
Anointed - The word in Greek “chiro” is used 5 times in the New Testament and always refers to God to doing the anointing or
(Isa 11:2 KJV) And the spirit of the LORD shall rest upon him, the spirit of wisdom and understanding, the spirit of counsel and might, the spirit of knowledge and of the fear of the LORD; The anointing of Jesus by the Holy Ghost was a direct fulfillment of the prophecy of Isaiah 11:2. He was also endued with mighty power to perform miracles. Jesus then being empowered of God had done good works within the confines of His ministry. He had performed healing on people as this was probably well-known and Jesus also cast out devils from people such as the Gadarene demoniac in Mark 5. These signs and wonders had confirmed that Jesus was of God and did nothing of His own. (John 8:29 KJV) And he that sent me is with me: the Father hath not left me alone; for I do always those things that please him. It is also interesting that Peter uses the term “Jesus of Nazareth” which was a term that was used as a term of derision but not meant in that way at this time. He obviously used it to state where Joseph and Mary lived, probably for the fact that there were many named “Jesus” which is the Greek name for “Joshua.” He wanted to make sure they knew which Jesus he was speaking of.
Acts 10:39 (KJB)
And we are witnesses of all things which he did both in the land of the Jews, and in Jerusalem; whom they slew and hanged on a tree:
Since Peter had been with Jesus from the beginning, he was a first hand witness to all that the Lord Jesus did. (John 15:27 KJV) And ye also shall bear witness, because ye have been with me from the beginning. Peter is now fulfilling the prophecy of John 15:27 as one who would be a witness for the Lord Jesus Christ. He was not only a witness to all the miracles Jesus did but also to the fact that He was crucified on a cross and was indeed dead. The word for “slew” in the Greek speaks specifically of the cessation of a person’s earthly life. The term “tree” denotes something made of wood which was the cross.
Acts 10:40 (KJB)
Him God raised up the third day, and showed him openly;
Openly - Manifest or visible
In the previous verse Peter makes a point that Jesus was definitely dead because there were many lies being told that He did not really die. Here he states that God raised Him up from the dead on the third day which Jesus had prophesied to His disciples. (Mat 16:21 KJV) From that time forth began Jesus to show unto his disciples, how that he must go unto Jerusalem, and suffer many things of the elders and chief priests and scribes, and be killed, and be raised again the third day.  Then after Jesus was raised from the dead, He was to be visibly seen by many people that He did rise bodily and not spiritually from the grave.
Acts 10:41 (KJB)
Not to all the people, but unto witnesses chosen before of God, even to us, who did eat and drink with him after he rose from the dead.
Chosen - Choose or designate beforehand
When Jesus rose from the dead, he had made Himself known to only certain witnesses which God chose before hand. Jesus did not go back into Jerusalem or the other cities and walk down the main street. He was made manifest to a certain group of chosen witnesses. This shows that there is nothing random in the Kingdom of God like those who believe in free will would have us believe. Just as God chose from the foundation of the world those who were to be saved, He also chose the witnesses to whom Jesus was going to show Himself. Peter was one who had spent the last 40 days on earth with the Lord and had eaten with Him which meant He rose bodily and was fully alive plus Peter was part of the chosen witnesses.
Acts 10:42 (KJB)
And he commanded us to preach unto the people, and to testify that it is he which was ordained of God to be the Judge of quick and dead.
Preach - Proclaim or herald
Testify - Testify fully
Quick - The living
Right before the Lord Jesus Christ went back to Heaven, He gave the disciples their future ministries. (Acts 1:8 KJV) But ye shall receive power, after that the Holy Ghost is come upon you: and ye shall be witnesses unto me both in Jerusalem, and in all Judaea, and in Samaria, and unto the uttermost part of the earth. These were the last words the Lord Jesus spoke on earth because when He finished them, He was taken up into Heaven. The disciples were given the task to preach the Gospel to the entire world and to fully testify, that is, they were to leave nothing out of their Gospel presentation. (Acts 20:27 KJV) For I have not shunned to declare unto you all the counsel of God. This is why the Apostle Paul was always getting into trouble because his testimony was not like today, “God loves you” or “Jesus loves you” and that is the extent. God and Jesus only love their Elect and not the unbeliever. (Psa 5:5 KJV) The foolish shall not stand in thy sight: thou hatest all workers of iniquity. Our testimony to the Gospel must contain all aspects of it and nothing is to be omitted so we will not offend anyone. The Gospel is an offense to the ways of the world. We must declare all the counsel of God and that includes the teachings of sin and hell. There also needs to be a full testimony to the fact that the Lord Jesus Christ is going to be the judge of all mankind. (John 5:22 KJV) For the Father judgeth no man, but hath committed all judgment unto the Son: When the Lord returns on the last day, He is going to judge all those unsaved who have died and all those who will be living at that time. (Mat 25:31-32 KJV) When the Son of man shall come in his glory, and all the holy angels with him, then shall he sit upon the throne of his glory: {32} And before him shall be gathered all nations: and he shall separate them one from another, as a shepherd divideth his sheep from the goats: The true believer who is born again in Christ passes from death to life and because of Calvary, will never come into judgment. (John 5:24 KJV) Verily, verily, I say unto you, He that heareth my word, and believeth on him that sent me, hath everlasting life, and shall not come into condemnation; but is passed from death unto life.