This answer is about the betting system. (There is also a dry cleaning process by the same name.)
A gambler "martinizes" by making a bet of x dollars on a bet that pays even odds. If he wins he pockets the money and the process ends. However, if he loses he next wagers 2x. If he wins this second bet he will be still "up" x dollars. If he also loses this bet he next wagers 4x, then 8x , etc.
The theory is that by continuing to double subsequent bets the gambler will eventually win back all the previous losses plus an extra x dollars.
This system doesn't work in the long run because the gambler will eventually be prevented by making the necessary last bet either because he runs out of money or hits the table limit.