They celebrated it with a grand community feast wherein the friendly native Americans were also invited. It was kind of a harvest feast, the Pilgrims used to have in England. The recipes entail "corn" (wheat, by the Pilgrims usage of the word), Indian corn, barley, pumpkins and peas, "fowl" (specially "waterfowl"), deer, fish. And yes, of course the yummy wild turkey.
However, the third year was real bad when the corns got damaged. Pilgrim Governor William Bradford ordered a day of fasting and prayer, and rain happened to follow soon. To celebrate - November 29th of that year was proclaimed a day of thanksgiving. This date is believed to be the real beginning of the present Thanksgiving Day.
Though the Thanksgiving Day is presently celebrated on the fourth Thursday of every November. This date was set by President Franklin D. Roosevelt in 1939 (approved by Congress in 1941). Earlier it was the last Thursday in November as was designated by the former President Abraham Lincoln. But sometimes the last Thursday would turn out to be the fifth Thursday of the month. This falls too close to the Christmas, leaving the businesses even less than a month's time to cope up with the two big festivals. Hence the change.