Monastiraki square is my favorite square in all of Athens. This is truly the place where many aspects of Greek culture mix together. It is located on the boarder of a very touristy area the Plaka and a very local area Psiri. It's also in an area filled with popular hot spots for the hip young Greeks. This square is a great place to just sit and watch many different aspects of Athens go by. You'll see the tackiest tourist, the hip Greek culture, and even glimpse a few of the sketchier aspects of this city in one place.
The shopping in this area is varied as well. Each street offers different choices because it is such a hub. Down one street you can buy all the souvenirs you need to bring home, down another you can buy various fashions, another offers Greek antiques, and don't forget about the many restaurants in and around the square. You can enjoy the cities best gyros, sip coffee with a view of the Acropolis, and even find a few non-Greek food choices.
There are also many sites to see in and around this square. There’s the Roman library, the Agora, and the Tower of the Winds all close to the square. But best of all there are glass floors all around the square that look down on an ancient river (now just a trickle) that used to run through this area. If you want a better look at what’s bellow, venture down into the subway station.
This square is really in the hub of modern Athenian life. Best of all, it's easy to find, since it has its own subway station "Monastiraki".