Which swear words or subjects are considered as most offensive in Russian, and why?
Minding the comment I had left to the question, I can give a simple generic answer (provided sensitive women and children had left the room): the most offensive words would those describing such tabooed things like the sex organs, and the sex act itself.
Of the first category the most offensive forms would be хуй (penis) and пизда (vagina), with former being so offensive that even its euphemism — хер (from the name of Old-Russian letter хѣръ which is today’s ха — basically an “Х-word”) — nowadays is just slightly less offensive.
And for the second it would be the verb ебать which means “to fuck” which is contraction of “to fuck your mother” meaning, in a sense, “How dare you argue with me? I’ve had an intercourse with your mother, and for all we know, you well may be my son, so behave accordingly!”. That explanation is culturally universal, really. Or so I’ve read.
I would also add that this kind of swearing is offensive by the virtue of its shock value, not necessarily literally. Of course, there are elements in society where this language is used in its own right, as a sort of group identification (such as criminal circles). But in a mainstream culture it is considered extremely crass and inappropriate.
Certainly, there would be traditional cultures where mention of one’s mother in any context other than respectful would constitute a deathly offence, perhaps, even within the borders of Russia and former Soviet republics. But for your average urbanite cosmopolitan Russian, so to speak, it would be offensive as it is plainly rude.