If you're looking for the classic Greek sweet treat, the first place you should check out is a bakery. There are thousands of bakeries in Athens all with different levels of quality and selection. The best way to pick a great bakery is to first check the window, and secondly check behind the counter. If you find a window full of tasty treats and people in the back actually baking, you're in for a fine treat.
Constantinides is a great bakery chain that never disappoints, with many locations all over Athens. Your next best bet for a great sweet treat is to sample some of the local chocolate. The problem you'll find in Athens is finding a truly Greek chocolate shop. Most chocolate shops make chocolates in the styles of Belgian chocolatiers. This isn't a bad thing most of the time, but it isn't a true Greek experience.
Max Perry, a local chain of chocolate shops has a decidedly more Greek approach to their chocolates, often blending with Greek cookies and fillings, much like the baked goods found at many of the bakeries. While Athens itself doesn't really have one particular sweet treat to call its own, many bakeries do sell some of the specialties from other regions in Greece. The Trigona from Thessaloniki is great, and Farsala Halvas, from Farsala is really unique. There are also some great seasonal treats like the Christmas cookies, and chocolate covered Easter bread that are well worth trying.
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