Sick leave or a sick day was time off from work granted by one's employer so that one could receive medical care and/or recuperate from one's condition.
In 2372, Rom came into Deep Space 9's infirmary with a severe ear infection after fainting on the job. Doctor Julian Bashir was aghast when Rom told him that he was not allowed to leave work during working hours, even for illness.
As a result of this incident, one of the demands from striking employees of Quark's was paid sick leave. Quark eventually granted their requests to end to work stoppage. (DS9: "Bar Association")