A work used to describe one who is "fucking ugly." This does not mean that the particular person is have sexual intercourse with a person who is ugly, but rather that the person's appearance is (in their opinion) ugly.
When the term fugly is used, it is obviously more extreme than the word ugly, and therefore cannot have conjunctions to euphemize or lesser the extent of the description. For example, one cannot say "man, that dude is a bit fugly" because really what is being said is that the dude is "a bit fucking ugly."
Cat: piss doodles
Dog: I find that offensive
Cat: I find your odor offensive
Dog: I find the word fugly offensive