I believe that the word for the wood being used to burn people (in the Middle Ages or during the Inquisition) got transposed onto some of the people they were burning (for being homosexual). Similarly the resemblance of the cigarettes to the kindling got the name transposed there as well.
1)
fag (British for 'cigarette') is a shortened form of 'fag-end', meaning the 'last piece' of something. It was used originally to refer only to the unsmoked portion of the cigarette, but over time came to refer to the entire cigarette. It has
no shared etymological ancestry with the term 'faggot'.
2)
fag/faggot (American for 'homosexual') has often been
claimed in recent years to derive from the burning of homosexuals on pires of faggots.
However, this suggestion is
almost certainly spurious, since this method of 'punishing' homosexuality was seldom employed, and died out (in Europe) centuries before the word 'faggot' came into usage to refer to homosexuals. It is more likely derived from 'faggot-gatherer', which was viewed in the nineteenth century as a women's profession that was common especially amongst older widows; from here, 'faggot' was indeed used in nineteenth-century British slang in a similar way to 'baggage', to refer to a shrewish woman. Early US literary references to 'faggots' from the 1910s and 1920s use the word in conjunction with other 'feminized' epithets for gay men, including 'sissies', 'fairies', 'queens', etc., lending further weight to the suggestion that this may be the etymological origin of the word.
Further online links which concur with the etymologies as per the above outline (
which have been discussed at length in sexological articles also):
--->
Straight Dope Staff Report: How did "faggot" get to mean "male homosexual"?
--->
http://en.wikipedia.org/wiki/Faggot_(epithet)