Some hotels will get an extra point and be designated "C'" which is a mid-grade between "C" and "B".
Deluxe - "L" Class - "Large rooms and public areas tastefully decorated and furnished". Includes dining facilities.
First Class - "A" - "Comfortably sized rooms and public areas decorated with pleasant furnishings". Also includes dining facilities.
Second Class - "B" - "Superior tourist class hotels." Includes dining facilities.
Third Class - "C" - "Modestly furnished. Adequate sleeping accommodations."
Fourth Class - "D" and Fifth Class - "E" - "Usually pre-war hotels, essentially furnished with bathroom facilities on a share basis."
I do feel the need to speak up in the defense of "D" and "E" class hotels. While the bathrooms may be shared, (though most will have some private baths), some of these older hotels have their own special charms and will offer more atmosphere than the shiny modern ones. And I've had wonderful luck with "C"-class hotels - my experiences have actually been less reliable in the supposedly-better "B"-class hotels.
Hotels in the first three classes are allowed to require half-board, particularly during high season. This will be an extra charge per person, and can be substantial. In practice, this is usually made optional.
My experience with hotel breakfasts in Greece is that they are as overpriced as hotel breakfasts anywhere -but they are convenient and may be the only option on a heavy day of touring. (You can find out my preferences on this subject in my article "Good Morning, Greece")
Compare some Athenian and other hotels on-line by visiting Hotels in Greece.
Good luck on your travels - let me know your experiences!
Compare airfares to Greece here: Find Greek Airfares

