The first (American) Thanksgiving occured in 1621 and was a harvest feast between the Puritan Pilgrims who had left England (fleeing religious prosecution) for the colonies and arrived in Plymouth, Massachussetts in 1620 and some of the local Native Americans.
Over the winter of 1620-1621, many of the Pilgrim settlers had perished due to cold, illness, and lack of food. The Native Americans had helped the English settlers with some planting, hunting, etc. during the spring and summer of 1621 and the two groups had a joint celebration in the fall with feasting and games.
A modern Thanksgiving celebration tends to be a large feast with friends and family. Foods served often include: Turkey (native to the Northeastern U.S.) and sometimes Ham, Gravy, Mashed Potatoes, Sweet Potatoes or Yams, Stuffing, Bread, Vegetables, Pumpkin Pie and other pies/cakes, Wine, etc. Families often contribute jointly to the feast, eat off the nice china, and then eat until the buttons are ready to pop off their pants.
Before and after dinner, men often watch Football (American style) on t.v. or might play sports outside; kids are running around playing with each other; and often families play cards or boardgames or watch a movie together in the evening/afternoon. Generally, it's either a wonderful visit or your family is driving you nuts -- depends on your family, I guess.
Here's a good website I found about the Pilgrims:
http://www.plimoth.org/learn/history/colonists/PilgrimPeople.asp
Edited to add: For the "Thanksgiving" aspect of Thanksgiving, the meal is often kicked off by saying a prayer of thanks-giving, where one or more members of the family give thanks for the good things in their lives. Honestly, with us and the ILs, Thanksgiving is the only meal all year where we say "Grace".