Isaiah describes for us the Messiah in His position as the Suffering Servant. Isaiah 53 actually contains several individual prophecies that are presented to us at one time.
When Jesus came, suffered for us and died on the cross, this prophecy was fulfilled. When He was pierced with nails in His hands as He was nailed to the cross for our transgressions and iniquities, this prophecy was fulfilled. The chastisement of our sins that fell upon Him allows our well being and healing.
Since there is more than one prophecy involved here a table has been created to help identify where the fulfillment occurred for each prophecy. For more on this and other prophecies by Isaiah check out the Isaiah Prophecy Fulfillment Table.
For a more in depth look at the entire 53rd chapter of Isaiah (Only 12 Verses) you can check out the light commentary that has been put together. Read Isaiah 53 Commentary.
Many bible commentators believe that Isaiah 53 actually begins 3 verses earlier with Isaiah 52 verse 13. (The oldest versions of the Bible do not contain chapter breaks). Some Jewish Literature, such as Midrash Tanchuma and Yalkut Schimeon (ascribed to Rabbi Simeon Kara, 12th Century), both see reference to the Messiah in Isaiah 52:13.
This chapter is sometimes referred to as the "great passional."
There are entire books written on this one chapter in Isaiah. Viewers of this page are encouraged to do independent study.
Articles Related To Messiah's Death Include:
Lubavitchers: Most Recently, a sect of Ultra Orthodox (Lubavitchers) in New York believed (and still believe) that Chief Rabbi Menachem Schneerson was the Messiah. When he died in 1994, they turned to Isaiah 53 to predict his resurrection and ascension to the throne of Jerusalem. To this day, this sect has not chosen another Chief Rabbi as they await Schneerson's resurrection.(1) However, this is not the opinion of all Lubavitchers.
1). Source: Discovery Series, The Jewish tradition of two rabbis. (RBC Ministries)