From:  http://www.bbc.co.uk/worldservice/learningenglish/language/2014/02/140204_todays_phrase_under_the_table.shtml 

Under the table

A giant table and chair
Artist Giancarlo Neri's work 'The Writer' consists of a 10m-high chair and 7.30m table, made of wood and steel. Photo: BBC.

Today's Phrase

Under the table is a phrase used to describe secretive behaviour - often suggesting corruption or illegality.
Examples:
If you want to open a business in a country where there is a lot of corruption, you may have to make under the table payments to officials.
Tony was a terrible driver. He kept failing his driving test - and eventually he got his driving licence under the table.

Take note

To drink someone under the table means to drink more alcohol than somebody else without becoming as drunk as they do.
Example:
Johnny was a heavy drinker. He could drink anyone under the table.

Interesting fact

The installation 'The Writer' was shown in Rome's Villa Ada in 2003, then in London's Hampstead Heath in 2005. It is now on permanent display at the Villa Reale in Monza, near Milan.