Did the disciples believe that Jesus was God or a prophet? I presume that their opinion changed when he died. They called Jesus lord, however lord is a somewhat common title back then.
Matthew 16:16 is perhaps the most prominent answer to this question.
So, the disciples became convinced by the miracles, the teachings, and the character of Jesus through His ministry that Jesus was the very Son of God.
Yet, it wasn't when Jesus died that their belief and understanding became solidified, but after His resurrection. The disciples were fearful prior to that. In fact, many were hiding while He was being crucified. All fled at His arrest, even though Peter made one valiant act in His defense.
If Jesus had only died, Christianity would not exist. It was the resurrection that turned "unschooled, ordinary men" into bold witnesses of Jesus who willingly faced persecution and death for their teaching that Jesus is Lord.
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:
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:
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:
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:
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):
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.