How does Jesus fulfill the offices of Prophet, Priest and King? I have heard that Jesus fulfills the three Jewish "offices" of Prophet, Priest, and King.  What is significant about this, and how specifically does or did Jesus fulfill those offices?
Prophet: In the old testament a prophet was typically one that communed with God to get revelation for the Israelites and was head of the 'church.'
I.E. Moses


  Exodus 33:11 And the Lord spake unto Moses face to face, as a man speaketh unto his friend. And he turned again into the camp: but his servant Joshua, the son of Nun, a young man, departed not out of the tabernacle.


Since Christ was head of the church while he was here on earth and received all the revelation from God during that time and passed it on to the 12 apostles he then would have the title of prophet.

King: One of the reasons that Christ was actually rejected by the Jews was because the Messiah of the old testament was supposed to be King of the Jews and come in Glory and overthrow those that oppressed the Israelite nation. Christ was (and is) the Messiah but the scriptures talked of 2 comings of the Messiah one in meekness (as the Son of Man) and one in Glory (his second coming see Malachi 3,4 Also see the whole book of Isaiah. Isaiah talks about both the first and second comings of the Lord and it can be confusing at times to know which he is talking about. Isaiah is also the one who most often refers to Christ as a King or descendant of kings:


  Isaiah 9:7 Of the increase of his government and peace there shall be no end, upon the throne of David, and upon his kingdom, to order it, and to establish it with judgment and with justice from henceforth even forever. The zeal of the Lord of hosts will perform this.


Referring to the fact that Jesus through His mother Mary and His adopted father Joseph He is descended from King David. The Jews, of course, were looking for a mighty descendant of David to overthrow the Roman rule of the time.

Priest: Priest has a lot of meanings depending on the denomination, but they often have duties to minister unto the members of a church. If you look at the life of Christ then it is easy to see that He fits this definition of a Priest.