A "farm" is a facility constructed by the Cylons in an effort to solve their difficulties with sexual reproduction using both male and female human test subjects.[1] There are multiple farms, with a notable one existing on occupied Caprica somewhere near Delphi.
Such facilities are overseen by the Fours, although Number One doesn't agree with the religious tract that the research has taken on to the Cylons ("The Plan", "No Exit").
Kara Thrace and Sue-Shaun are taken to a farm (formerly the Delphi Convalescent Institute) after being wounded in an ambush orchestrated by Caprica-Cavil ("The Farm", "The Plan"). Inside, the farm held hundreds of women hooked up to medical equipment, used as living incubators. Thrace is cared for separately (and under the guise of mending from her injuries) and her case reviewed by a Four using his human name, Simon.
According to Sharon Agathon neé Valerii, the Farms haven't produced viable Human/Cylon hybrid offspring, ergo the experiment that manipulates Karl "Helo" Agathon into falling in love with Valerii. Some Cylons theorize that the farms failed because a child must be conceived in love (keeping in mind that the Cylons believe that "God is love"), and as a result Caprica-Valerii carries the first and only successful hybrid pregnancy.
After Thrace escapes the farm where she is held, Caprica Resistance leader Samuel Anders promises her that his band of fighters would attempt to destroy as many of the remaining farms as they could (TRS: "The Farm") before they are rescued months later (Lay Down Your Burdens, Part I).
The farms are abandoned after the Cylons decide to leave the Colonies (TRS: "Lay Down Your Burdens, Part II"). It remains unknown how the captured humans are dealt with as a result of this decision.