Where does the term "three sheets to the wind" originate from?

Sheet is the nautical term that refers to the rope used to secure a ship's sail.
"Three sheets to the wind", or "three sheets in the wind", compare the way the sails (not anymore secured) are moved from the wind, to the way a drunk person walks on the street.

First, it means “being inebriated” or “completely drunk”.

Second, regarding etymology, urbandictionary.com says:

origin: sheets actually refer to the ropes that are used to secure a ship's sail. If the 3 ropes used were loose in the wind, the sail would flop around, causing the ship to wobble around, much like a drunk.

Apparently, variants exist, as the New Oxford American Dictionary has:

two (or three) sheets to the wind: (informal) drunk.

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Regarding the etymology of sheet in this meaning, NOAD agains comes to the rescue:

Old English scēata [lower corner of a sail] of Germanic origin; related to Old Norse skauti (kerchief)

I think there is a significant difference between "to" the wind and "in" the wind. "To" meaning toward the wind (nautical) and "in" literally meaning out in the wind (windmill).

• To the wind - A sheet onboard secures the sail to the vessel from the clew (loose corner of the sail), it is fastened to catch the wind, as the boat comes about it is loosened and the sheet on the opposing side is fastened, this keeps the sail in proper position to catch the wind. Having three sheets "to" (toward) the wind, may very well mean a vessel with its sails in the completely wrong position, literally fastened up wind (the wrong side).

• In the wind - A windmill, is completely unbalanced with one or three sheets "in" the wind, as opposed to two or four. You would never fly three sheets in the wind, you must always balance the sheets.

John Bee, Slang. A Dictionary of the Turf, the Ring, the Chase, the Pit, of Bon-ton, and the Varieties of Life (1823) has this early glossary entry for the "three sheets in the wind" form of the expression:

Three-sheets in the wind.—Naval, but naturalized ashore, and means drunk, but capable of going along—like a ship which has three sheets braced—main, mizzen, and foresail.

The earliest Google Books instance of this variant appears in "Travelers in America," in Niles' Weekly Register (May 2, 1812), quoting Thomas Ashe, of Great Britain, in connection with his travels to the U.S. state of Kentucky:

"The inhabitants of Louisville are universally addicted to gambling and drinking." Their mode of living may be taken from the following instance, which "conveys a general idea of the whole state." Mr. A. had met with a Kentuckyan on the road, and accompanied him home. "On entering the house, which was a log one, fitted up very well, the Kentuckyan never exchanged a word with his wife or his children, notwithstanding he had been absent several days. No tender inquiry, no affection or sentiment, but a contemptuous silence, and a stern brutality, which block up tall the avenues to the heart. The poor woman, whom I pitied, (for it is a bet, that the women do not degenerate in proportion to the men, but continue to this day amiable) made a large bowl of drink called toddy, composed of sugar water, whiskey, and peach-juice, and handed it to her husband with all the servility of a menial ; he drank and handed it to me, who followed his example, and found the liquor excellent." It must not be wondered at that the poor, untutored, savage Kentuckyan, got "more than two thirds drunk," that is, as the sailors term it, three sheets in the wind, and the fourth shivering, before the dinner was ended, upon a liquor which this great man found excellent.

The instance cited in Callithumpian's answer, which appears to be from September 26, 1813, also describes experinces in Kentucky; however, the locale seems merely coincidental, given that both Thomas Ashe and Bishop Asbury were travelers from elsewhere, just passing through the state.

Another early instance appears in "How Lorenzo Raised the Devil," in the [Leesburg, Virginia] Genius of Liberty (August 26, 1817):

The woman opened the door and received her husband with as much tenderness as surprise. He was about three sheets in the wind, that is to say a little intoxicated and began to talk loud and swear. She hushed him by informing him that a minister the famous Lorenzo Dow was asleep in the next room.

The earliest instance of the variant "X sheets to the wind" that I have been able to find comes from the [Pennsylvania State University] Free Lance (January 1, 1894):

While Mr. Hoard occupied the gubernatorial chair of the Badger state [Wisconsin] it chanced that he was passing through a street in Milwaukee late one night when he espied two old friends approaching, one of them three sheets to the wind and the other piloting him. "Brace up Jack" said the sober man, "here comes the governor."