A deal or a payment made “under the table” is one made secretly or covertly. The phrase “under the table” might come from card playing and gambling, but some liquor during prohibition (1920-1933) was served “under the table.” A bribe is often described as a payment made “under the table.” Workers are often paid “under the table” (or “off the books”) to avoid paying taxes.
 
“Under the table” has been cited in print since at least 1938. A 1938 book described “Under the table, something given as a bribe.” In April 1938, Congress passed a law so quietly and quickly it was said to have been passed “under the table.” In May 1938, the lieutenant governor of Pennsylvania described a contractor deal by “men whose only understanding of a deal is one under the table.”
 
Other “table” phrases include “bring to the table” (to contribute something), “on the table” (something negotiable or for discussion) and “off the table” (something non-negotiable).
     
 
Wiktionary: Under the table
Alternative forms
under-the-table (attributive use)
Prepositional phrase
under the table
1. (adjectival and adverbial) Secret(ly), covert(ly), undisclosed(ly).
The payment was under the table.
I was paid under the table, so I didn’t have to pay taxes on the income but I couldn’t complain when I was cheated.
2. Used other than as an idiom: see under,‎ table.
 
(Oxford English Dictionary)
under the table
In a clandestine or underhand way; secretly; illicitly. Freq. attrib. (usu. hyphenated): secret, hidden, done in a clandestine or illicit way. Cf. under the counter at counter n.3 4b.
1938 F. D. Sharpe Sharpe of Flying Squad 334   Under the table, something given as a bribe.
1943 H. A. Wallace Cent. of Common Man (1944) 86   We can create co-operation or conflict; unity of purpose or under-the-table dealing.
1952 J. Lait & L. Mortimer U.S.A. Confidential ii. xi. 92   Liquor licenses cost anywhere from $10,000 to $50,000 under the table.
   
7 April 1938, The Plain Dealer (Cleveland, OH), “The News Behind Today’s News” by Paul Mallon, pg. 6, col. 3:
Slipped Through
WASHINGTON, April 6.—That billion and a half dollar RFC lending authorization was slipped through Congress so fast the legislators are now saying it must have been passed under the table.
 
4 May 1938, The Morning Herald (Uniontown, PA), “Kennedy Lashes Contractor Deal,“pg. 9, col. 3:
“We haven’t had a New Deal,” he declared, “we’ve had a contractor deal, a deal by men whose only understanding of a deal is one under the table.”
(Lieutenant Governor Thomas Kennedy—ed.)
 
23 June 1938, Augusta (GA) Chronicle, “Mayor Burnet R. Maybank To Seek S.C. Development” (AP), pg. 5, col. 3:
Neville Bennett, chairman of the Ways & Means Committee, said the state was not spending money “under the table.”
 
30 June 1943, Register Star-News (Sandusky, OH), pg. 6, col. 3:
But money is paid “under the table”, according to the OPA.
(Office of Price Administration—ed.)
 
20 November 1943, Wisconsin State Journal (Madison, WI), “Roundy Says,” pg. 4, col. 1:
But the boys claim they seen Beggs over there slipping him a little money under the table.
 
5 September 1946, Southeast Economist (Chicago, IL), “A Penny for Your Thoughts,” pg. 1, col. 6:
Frances Frola, 2840 Komensky ave., factory worker: “It makes me sick to see some people so piggish they won’t give men and women who fought for them a place to live without holding them up. It’s a pretty bad situation when you have to slip money under the table for things you want and need which are in the ‘back room’ but aren’t for sale unless you dangle a big fat bonus in front of some one’s face.”
 
21 December 1946, San Mateo (CA) Times, “Vets Fight to Build Pre-Fabs,” pg. 2, col. 5:
It was disclosed to the county planners that Thorne and Davis had “made up their minds” that they would prove it is possible for veterans to build their homes under government provisions without “passing money under the table to a contractor” because of the former price ceiling restrictions.
 
30 June 1950, Dallas (TX) Morning News, “Prosecution Soft Pedals Tax Agents’ Racketeering” by Drew Pearson, pt. 3, pg. 5, col. 2:
Almost simultaneously with the Kenmoor shakedown, John Galgano, another investigating agent, uncovered a $300 check paid under-the-table to the Sherman Chevrolet Company, 561 Central Park Avenue, Yonkers, N.Y., as a bonus on the purchase of a new car.