Guinness is a stout.
Stout is made of water, hops, yeast and roasted barley, hence it is darker in colour than most beers, black or dark brown. It seems to be a name for porter, a strong beer drunk by the boatmen of London, although stout seems to predate that. I'm sure some sources will say porter is a kind of stout. I've not drunk much stout before this last year - Guinness in High Wycombe, Murphys on occasion - but there are some interesting one around in a very of styles.
Breakfast stout seems to be a stout with the sort of ingredients added that you'd have for breakfast - not bacon and eggs but oats and coffee and (how French!) chocolate. I've had oatmeal stouts and milk stouts, but Anarchy Brew Co's Sublime Chaos is the first breakfast stout I've had. The coffee and oat flavours come through, and moderate the burnt flavour; it also controls the sweetness I suspect and I gather that an earlier attempt with a different bean was too sweet. This has a back of the throat flavour, somewhat warming, very nice without being great.
Examples: Anarchy Brew Co Sublime Chaos.