Plaza de Olavide
Again a enjoyable place located in my neighborhood, Chamberi. We go there a lot, specially on weekends. When my daughter came to visit, she took some pictures of dogs and children playing. Later on I used the pictures she took as a reference photos. This is an all-in one plaza for me: coffee-shops, a jazz club, bars, restaurants, an ice cream parlor, a drugstore and a hotel. There is also a playground for children and a tiny bookstore in the center of the plaza which specializes in used books and vintage post cards and pictures of Madrid.
What I love about it is that, on one hand, it is your neighborhood plaza. Not just the place to sit for a drink or to have brunch, but also where people take their children after school, or where they walk their dogs while reading the newspaper. And on the other hand, Plaza de Olavide is the starting point of a night out on the town because many cool clubs and restaurants are in this same area. It has it all, daytime activities and nighttime fun.
Its location also helps to make it even more charming, because is at a walking distance from Gran Vía and very close to Plaza Colón. Also pedestrian street Fuencarral, filled with all kinds of shops, is 10 min away. It would seem busy and noisy at times, but it could also feel quiet and calm at certain hours. It depends on what you want to do, but it is definitely worth a visit.