Did Jesus' disciples believe Jesus was God or a prophet?

While it is true the disciples were often confused and varied in levels of faith, enough is implied in the gospels to give evidence that they were impressed by the Holy Spirit at times to believe that He is God.

Matthew 14:30-33 (NKJV) But when he [Peter] saw that the wind was boisterous, he was afraid; and beginning to sink he cried out, saying, “Lord, save me!”. And immediately Jesus stretched out His hand and caught him, and said to him, “O you of little faith, why did you doubt?” And when they got into the boat, the wind ceased.

Then those who were in the boat came and worshiped Him, saying, “Truly You are the Son of God.”

Jesus walked on water, Jesus save Peter, but the emphasis in the above passage is that the disciples worshiped Him. They would not have broken the ten commandments and worshiped Him if they did not think Him as God. In addition they called Him the Son of God, not a son of God.

John 6:66-69 From that time many of His disciples went back and walked with Him no more. Then Jesus said to the twelve, “Do you also want to go away?”

But Simon Peter answered Him, “Lord, to whom shall we go? You have the words of eternal life. Also we have come to believe and know that You are the Christ, the Son of the living God.”

Jesus spoke with authority that is not from this world. When He preached “I am the bread which came down from heaven”, many disciples who thought of Him as the Messiah (according to the testimony of John the Baptist) left, because Jesus was implying that He was much more.

This was a hard teaching, but the twelve stayed because the Holy Spirit worked in them to believe. In addition, they still stayed when Jesus said "Before Abraham I am" and "My Father and I are one", despite others wanting to stone Him for blasphemy.

Matthew 17:2 He was transfigured before them. His face shone like the sun, and His clothes became as white as the light.

Matthew 17:5 suddenly a voice came out of the cloud, saying,“This is My beloved Son, in whom I am well pleased. Hear Him!”

Peter, James and John saw Jesus transfigured in His glory! Would they have thought He was a mere saint? If that was not enough, they then heard the voice of God testify that "This is My beloved Son" If it did not click for them, it would have clicked right there. Jesus is not merely a son of God, but the living Son of God in the fullest literal sense.

Maybe this is what Dan Andrews was trying to ask (I could be wrong), "did the disciples believed Jesus was God ONLY after the resurrection or was it before drawing from the statement in Mathew 16:16 as Narnian was pointing out?

To answer that question we would have to know what Peter meant by "Son of God" - there are at least two possibilities:

A) Peter believed Jesus was the Messiah prophesied to save Israel who was appointed by / from God or;

B) Peter believed Jesus was the Messiah prophesied and was God himself in the flesh

the choice, I would imagine, would need to be decided by looking at the Greek and the context of this statement in the Gospel of Matthew.

The disciple John most certainly believed Jesus was God. He wrote in John 1:1 "In the beginning was the Word, and the Word was with God, and the Word was God. Then in verse 14 he told us "the Word became flesh."

Additionally he records the words of Jesus who said,

"I and the Father are one." (John 10:30.)

“the Father is in me and I am in the Father." (John 10:38.)

"Believe me that I am in the Father and the Father is in me," [ONE PERSON] (John 14:11)

If you want to tell me that the disciples did not believe Jesus, then that means they were atheists, and that is beyond the pale, even the crowds believed and so did Martha who said to Jesus “Yes, Lord, I believe that you are the Messiah, the Son of God, the one coming into the world.” John 11:27

The disciples were continually learning. They didn't have it all together and they didn't truly understand everything Jesus said or tried to teach them. They saw him calm a storm and raise the dead to life again. They even were given power to heal the sick and cast out demons, but we're limited by their lack of faith on occasion. Peter figured out Jesus was the Christ, the son of the Living God, but then he denied knowing Him. It wasn't until after the resurrection did the disciples eventually come to know who Jesus really was. Thomas got it right fairly quickly at least in word when he exclaimed "My Lord and my God".

In addition to the great answers here, I would like to answer the "being God" bit. When John wrote his gospel (a long time after Jesus' death and resurrection), he wrote these famous words (John 1:1):

In the beginning was the Word, and the Word was with God, and the Word was God.

Yes - they thought many things, and at various stages in His ministry, differing "circles" of His disciples expressed different things.

Of His closest disciples (those we generally associate with the term "disciple", aka The Twelve), Peter declared Jesus to be the Christ first:

Simon Peter answered, "You are the Christ, the Son of the living God." {Matthew 16:16}

This is the same Peter who went on to deny Jesus three times during His trial before His crucifixion.

It took Thomas seeing the resurrected Jesus to realize who He is:

"Unless I see in His hands the imprint of the nails, and put my finger into the place of the nails, and put my hand into His side, I will not believe." .. Then He said to Thomas, "Reach here with your finger, and see My hands; and reach here your hand and put it into My side; and do not be unbelieving, but believing." Thomas answered and said to Him, "My Lord and my God!" Jesus said to him, "Because you have seen Me, have you believed? Blessed are they who did not see, and yet believed." {John 20:24-29}

Certainly they recognized Jesus to be "important" early on (see how fast they respond to His calls of them), for example when they ask to learn how to pray:

It happened that while Jesus was praying in a certain place, after He had finished, one of His disciples said to Him, "Lord, teach us to pray just as John also taught his disciples." {Luke 11:1}

When each of the twelve became "aware" of who Christ was is not told to us - certainly it was by Pentecost, though.

Those not close to Christ (or who were only peripherally-aware of Him), did not know who He was - but worried about Him:

And King Herod heard of it, for His name had become well known; and people were saying, "John the Baptist has risen from the dead, and that is why these miraculous powers are at work in Him." But others were saying, "He is Elijah." And others were saying, "He is a prophet, like one of the prophets of old." But when Herod heard of it, he kept saying, "John, whom I beheaded, has risen!" {Mark 6:14-16}