This bar ("The Old King of Wines") was opened 100 years ago and hasn´t changed much since then. It specializes in sweet wines (they are served with a cookie - don't hesitate to dip it in the wine, that's what it's for!!), and also has some good tapas: albóndigas (meat balls), cheese, escabeches (pickles)... and the specialty: "torrijas", bread slices soaked in milk and fried, served with cinnammon and sugar/honey.
The beauty of the place also comes from the fact that it is not too crowded or too popular with "standard" tourists - probably because it is placed in a street with not much pedestrian traffic.
However, it is very easy to find and well within the "must see" area of Madrid: don´t trust the official address (La Paz street is tiny), look for at it in calle Bailén w/ calle Mayor, right opposite the new Almudena Cathedral (and, therefore, very near the Royal Palace).
While you're there, and after having that wine and that torrija, you might want to cross the street and have a look at the rests of the Arab walls of Madrid, they are only 40 yards away: go down the "Cuesta de la Vega" and you'll see them immediately behind the cathedral.