Thanksgiving is generally considered to have begun in either 1621 ir 1622 ( I don't recall which) after the Pilgirim's had a day of Thanksgiving to thank God for a good harvest (the first one had been rather poor, and IIRC about half the colony died).
There are only two primary sources for the Thanksgiving story. The first is from Governor William Bradford's history of the Plymouth Colony.
"They begane now to gather in ye small harvest they had, and to fitte up their houses and dwellings against winter, being well recovered in health & strenght, and had all things in good plenty; for some were thus imployed in affairs abroad, others were excersised in fishing, aboute codd, & bass, & other fish, of which yey tooke good store, of which every family had their portion. All ye somer ther was no wante. And now begane to come in store of foule, as winter aproached, of which this place did abound when they came first (but afterward decreased by degree). And besids water foule, ther was great store of wild Turkies, of which they took many, besids venison, &c. Besids they had aboute a peck a meale a weeke to a person, or now since harvest, Indean corne to yt proportion. Which made many afterwards write so largly of their plenty hear to their freinds in England, which were not fained, but true reports.
"William Bradford. "Bradford's History Of Plimoth Plantation." Boston: Wright & Potter Printing Co., State Printers... 1898. p. 127
The other is from Mort's Relation...by Edward Winslow
"Our Corne did proue well, & God be praysed, we had a good increase of Indian Corne, and our Barly indifferent good, but our Pease not worth the gathering, for we feared they were too late sowne, they came vp very well, and blossomed, but the Sunne parched them in the blossome; our harvest being gotten in, our Governour sent foure men on fowling, that so we might after a more speciall manner reioyce together, after we had gathered the fruit of our labors; they foure in one day killed as much fowle, as with a little helpe beside, served the Company almost a weeke, at which time amongst other Recreations, we exercised our Armes, many of the Indians coming amongst vs, and among the rest their greatest King Massasoyt, with some nintie men, whom for three dayes we entertained and feasted, and they went out and killed fiue Deere, which they brought to the Plantation and bestowed upon our Governour, and upon the Captaine, and others. And although it be not alwayes so plentifull, as it was at this time with vs, yet by the goodneses of God, we are so farre from want, that we often wish you partakers of our plenty."
Mourt's Relation: A Journal of the Pilgrims at Plymouth
George Morton, William Bradford, Edward Winslow, and Robert Cushman (Dwight B. Heath, ed.)
The American Tradition of having a day of Thanksgiving continued, not as a specific holiday, but as a way of giving thanks to God for specific events...(much the same as you see occasionally happening in the Bible). George Washington and (IIRC) the Continental Congress declared days of Thanksgiving on occaison, and as was said here President Lincoln issued a Proclimation of Thanksgiving during the Civil War (it was after a Union victory, Sharpsburg I think).
The Pilgrim tradition though continued in New England, and it was a yearly, though regional, holiday in New England for most of American History. From there it spread to the rest of the US as New Englanders moved to other parts of the USA (to get jobs, etc.) but still celebrated Thanksgiving. By the end of the 19th century it was pretty much a national holiday.
Why it is on a Thursday, I do not know.
It used to be a lot more about actually thanking God... some churches still have Thankgiving services you can go to.
Answered By: Larry R - 11/25/2007