Did Jesus' disciples believe Jesus was God or a prophet? 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" was a somewhat common title back then.
Matthew 16:16 is perhaps the most prominent answer to this question.


  Simon Peter replied, “You are the Christ, the Son of the living God.” 
  Matthew 16:16 ESV


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.