I stayed by Puerta del Sol, which is actually a historical building rather than a gate or a door. The plaza is always flooded with people trying to get to the metro station or taking in the sights. Madrid really loves its fountains, and the first fountain I saw was at Puerta del Sol. The plaza also has a statue of a bear climbing a tree, which I didn't understand but is all over the souvenirs. Apparently, it is the symbol of Madrid, in Spanish known as El Oso y El Madrono (the bear and the strawberry tree), which is linked to the city's name.
Plaza Mayor is also full of street performers (they are big on cartoon characters) and many cute outdoor restaurants. They may seem expensive but actually are decently priced for sitting outside in the sun, watching people trying to earn a buck for their talents (or lack of) and enjoying delicious paella with a glass of sangria.
Right next to the plaza, is El Mercado de San Miguel, an indoor food market. It's a great place to try the different foods Madrid has to offer. It's full of people eating tapas and enjoying a cold drink, and it's also a great way to meet locals and learn about the cuisine. The best part is that this market is open every day! I liked the food here so much that I went twice in one weekend.
El Retiro is a huge park in the eastern part of the city, that is almost comparable to Central Park. It has many fountains, lakes, a place to row boat and Palacio de Cristal. They say it's a crystal palace, but it is more like an empty room or greenhouse looking building. It is gorgeous however, and the inside gives a great view of the landscape of the park.
Museo del Prado is right next to El Retiro. I was really looking forward to seeing it but I don't think it was a museum for me. I like art museums that have all types of art, paintings, sculptures, decorative arts, jewelry, etc. Museo del Prado mostly just has paintings. Most of the paintings I found to be very dark and gloomy but they were interesting to see.
Madrid was a great experience and reminded me a lot of NYC. There were people selling fake DVDs, hand bags and watches everywhere. People were even selling water bottle for 1 Euro. And the street food was excellent. I really want to go back!