Syntagma square is the heart of modern Athens, The parliament building and the tomb of the unknown soldier dominate the one end of the square, while across the street the open air cafeterias and fountain contribute to a laid-back atmosphere.
Syntagma is also a major transportation hub. The metro station below is a central exchange point for two popular train lines. From here, the Olympeion and the National Gardens are withing a short walking distance, and a good place to start a walking tour of the historic center of Athens. It's a popular hub for Greeks who prefer the square benches and outdoor cafeterias for relaxation and people-watching (a popular activity in Greece), and a busy pass-through location for those who walk from the metro stations toward the trendy shopping street (Ermou) that starts at the square and ends at Monastiraki.
It is generally a clean and safe place to visit. The large hotels that frame the square are among the most historic and luxurious in Greece, being in close proximity to the seat of government and the multitude of embassies and consulates that surround the Parliament building.
It is also a place of every major protest by Greek citizens during demonstrations and strikes. The vast majority of them are peaceful and fall within the democratic expectations of citizen protest, but a small percent of them end up erupting in violence. Regrettably, these latter activities are also the ones promoted and sensationalized by world media.
The vast majority of visitors to Syntagma find it an ordinary and safe place to hang out, feed the pigeons, and photograph the picturesque Presidential guards and their traditional skirt-and-pom-pom-sporting uniforms.
Behind the Parliament building you can walk through the National Garden of Athens - a botanical garden that offers a quiet place to walk around and enjoy the relative quiet in the middle of the city. A few blocks to the south, you will find the imposing Zapeio center and a few quiet cafeterias, and if you keep walking south you will arrive at the Olympeion - a major archaeological site where you can see the imposing temple of Zeus. From there, or from Zapeio Megaro you can cross the very busy highway (Ardittou/Vasileos Konstantinou street) to visit the Panathinaiko Stadio which was the place where the first modern Olympics took place in 1896.