The informal phrase under the weather means: not completely well; slightly ill or depressed.

This phrase occurs, for example, in the following by the solo vocalist Shirley Ann Scott, published in The Stage (London, England) of Thursday 26^th October 2006 [page 12, column 4]:

A really important consideration is that unlike in a band, where if one member is a bit under the weather, the others are able to compensate, as a solo singer there’s no place to hide on stage so you have to make sure you stay in top condition at all times. That means it is up to you to do your vocal exercises and practise your routines regularly—there are no other people around to keep reminding you.

In the phrase under the weather, the image is of a ship caught in a storm. In fact, the noun weather has long been used to denote a storm—as in the following from the description of the Island of Tenerife, published in A Voyage to New Holland, &c. In the Year, 1699 (London: printed for James Knapton, 1703), by the English buccaneer and explorer William Dampier (1651-1715) [chapter 1, page 10]:

The Port is but ordinary at best, and is very bad when the N. W. Winds blow. These Norwesters give notice of their coming, by a great Sea that tumbles in on the Shore for some time before they come, and by a black Sky in the N. W. Upon these Signs Ships either get up their Anchors, or slip their Cables and put to Sea, and ply off and on till the Weather is over.

The earliest literal use of under the weather that I have found refers to a ship caught in a storm—it is from a correspondence from London, dated Tuesday 10^th January 1786, published in The British Chronicle. Or, Pugh’s Hereford Journal (Hereford, Herefordshire, England) of Thursday 12^th January 1786 [page 3, column 3]:

It is with the utmost concern we inform our readers of the unhappy fate of the Halsewell East-Indiaman, Richard Pierce, Esq. Commander, which was totally lost, on Friday last, off Peverel Point, on her outward-bound voyage to Bengal.
The Pilot had left the ship on Tuesday noon; after which she sailed down the channel with a fair wind till about four o’clock on Wednesday morning, when a very hard gale, with a heavy fall of snow, came on, by which the ship received so much damage as to admit six feet water into the hold. About eleven o’clock on Wednesday morning, when they thought they were between the Lizard and Start Points, they cut away the main and mizen masts, then wore ship, and endeavoured to make Portsmouth under jury-masts. They stood up the channel on Wednesday afternoon, and all the day on Thursday. In the afternoon of the last-mentioned day a heavy gale blew from the south, which by degrees drove them on a lee-shore, in spite of all their endeavours to avoid it. Between one and two o’clock on Friday morning they saw land, and came to anchor, at which they rode about an hour. But having either driven or parted, they then let go the only anchor left them, with which they were unable to bring up the ship, as the hurricane continued to increase. In this state the vessel drove upon the rocks at the Headland of St. Alban’s, about three leagues to the east of Portland, and in less than an hour was dashed to pieces.
Had they been fortunate enough to have driven clear of that head land, they would have got into Swanage or Strickland Bay, where they might have had safe anchorage under the weather.

The earliest figurative uses of the phrase under the weather that I have found are as follows, in chronological order:

1-: From a letter to the Editors, by a person signing themself ‘Union’, published in The Western Monitor (Lexington, Kentucky, USA) of Friday 31^st March 1815 [page 3, column 5]:

In two short months the scene is wholly changed. The whole machinery of commerce is ere this in that country repaired, and in no long time the rust it had contracted will be worn off, and its usefulness and beauty be totally restored. On such a restoration the liberal spirit of Kentucky would dwell with pleasure even if she herself were not to be benefitted. To this pleasure may be added the satisfaction such as a brother feels when by a sudden turn of things, a brother, who had been under the weather rises into usefulness and independence, able to stand of himself, and to impart as well as receive assistance.

2-: From Old Iron Sides. All Hands Ahoi!, published in the Aurora (Philadelphia, Pennsylvania, USA) of Friday 3^rd October 1817 [erroneously dated Friday 4^th October 1817; page 4, column 4]:

The landlord, with an eye to business, observed that there was a very honest fellow in the neighborhood, but a little under the weather or so, who had sailed in Old Ironsides, with Hull and Bainbridge—the reason why he was under the weather, was, that he was just recovering from his wounds.—and had a small ship’s crew of wife and children, which had left scarce a shot in the locker. *

* Here, the phrase a shot in the locker designates money in reserve.

3-: From The National Advocate (New York City, New York, USA) of Tuesday 9^th March 1819 [page 2, column 3]:

We have great pleasure in announcing the arrival of the Hon. Surrogate from Albany—in good health, but his spirits are somewhat dashed at the prospect of affairs. He knows who will be appointed Judge, but will not tell. He appears to be “under the weather.”

4-: From the New-York Daily Advertiser (New York City, New York, USA) of Wednesday 15^th November 1820 [page 2, columns 5 & 6]:

THEATRICAL.

We are pleased to notice, that the Managers of the Theatre, with a liberality that does them honour, have given the house tomorrow evening to the Benefit of Mr. JOHN BAPTIST, a worthy citizen of our city, whose mechanical ingenuity and taste in the building of Boats for strength, beauty and speed, has long been fully established. He is the builder of the “New-York,” which won the race on Saturday, and of the “Knickerbocker,” exhibited in the American Museum, which won a former race. Through misfortune and a large family Mr. Baptist has become reduced in a pecuniary point of view, & is now, as sailors say, “under the weather.” This fact we trust will be sufficient to call forth the liberality of the citizens of New-York, who we are proud to say, have ever been ready to lend a helping hand to their fellow-citizen in adversity. The entertainment for the evening will be highly pleasing to the friends of the drama. The Club of Knickerbockers, and the winning boat in the late race, will be present.
KNICKERBOCKER, Jun.