Ryokan is a type of traditional inn dating from the Edo period (1603–1868), when they served travelers along Japan's highways. They typically feature tatami-matted rooms, communal baths, and other public areas where visitors may wear yukata and talk with the owner.  The style of Ryokan in Tokyo varies wildly from an urban hostel, to a humble guestroom in someone's spacious home, to a more traditional style of inn/hotel.  Ryokans are consistent with their gracious hospitality, low price and exposure to traditional japanese culture.

Like a bed and breakfast, you can add daily breakfast to your bill for a nominal fee:

Breakfast: A japanese breakfast (see below) consists of miso soup, seaweed, small amount of fish, rice, and japanese pickles. 

Western breakfasts (like the one from the Ritz above) can sometimes be substituted but are rarely satisfying to the American traveler.  Eggs are scrambled into small curds and are typically the consistency of cottage cheese.  Bacon is pretty rare, so you'll probably get sausages (which look and taste exactly like mini hot dogs), orange juice is pricey and milk is rare as well.