A fag is either a cigarette, or a derog. expression for a homosexual.
It derives from the old public school era (because they are run by members of the public - what Americans refer to as private school, heaven knows why). A ‘fag’ was a junior school child who would serve a senior child (clean his shoes, do his washing, run errands, be whipped when out of line etc), all still pretty common place in some schools over here!