A dog-end is the small leftover of a cigarette after someone has smoked it — the stub, often with a bit of ash, that gets dropped or put out. This is British informal speech, a bit rough, the kind of word you hear in the street or in older writing about pubs and pavements, not in a careful school essay. American speakers are more likely to say butt. Though actually, even in Britain many people now just say stub or cigarette end; dog-end feels a little dated, and it can sound dirty because it names something thrown away. It is only this leftover, not the whole cigarette, and not a dog’s bottom.