Fasting is a period where one gives up food and usually most beverages other than water. It is often done (in many cases for religious reasons) to remind oneself of how fortunate one is.
In 2268, Garth of Izar explained to James T. Kirk and Spock that Governor Cory would not be eating with them, saying he was fasting. (TOS: "Whom Gods Destroy")
The Bajoran Time of Cleansing is a month-long occasion in the Bajoran religion where the people of Bajor fast. (DS9: "Bar Association")
Additionally, Bajorans fast to commemorate Ha'mara, or the arrival of the Emissary of the Prophets (Benjamin Sisko). (DS9: "Starship Down")
In Klingon mating rituals, before his wedding, the groom goes on the road to Kal'Hyah, a period of fasting and rigorous physical challenges, with his closest friends. (DS9: "You Are Cordially Invited")