Monastiraki is the old market in the historic center of Athens, located at the north foot of the Acropolis rock. It is the place just about every tourist will venture for a few hours every visit, and it's not a bad place to spend an afternoon. It's cobblestone, pedestrian streets are lined with open shops that sell everything from antiques and clothing, to music and tourist knick-knacks. It's usually the place where you can buy all your cheap souvenirs for everyone back home, can have lunch outdoors, or enjoy a drink watching the crowds go by and the Acropolis view from below.
As you exit the Monastiraki metro station you can orient yourself easily (use this Athens map for reference). With the Monastiraki station on your back, the street to your right, Areos, would take you up a steep path to the Acropolis. After the first hundred meters or so, the path becomes steep and devoid of shops. To your right you will enjoy a good view of the Agora of Athens, and if you venture to your left, you will get lost in the quiet streets of the old historic district of Plaka.
Again, with the Monastiraki station on your back and looking straight ahead across the little square, you will see the cobblestone, tourist shop-packed street that would take you all the way to Mitropoleos. This is Pandrosou Street.
While tourist shops are sprinkled throughout the area, this street alone specifically caters to the tourist souvenir-hunting crowds. There is something for everyone there. Replicas of ancient Greek statues (the vast of them exceptionally ugly, with a good portion approaching tasteless), leather goods, carpets and clothing (most of very good quality), and tons of jewelry, all scream from the open front shops to be considered for a trip to your home. Prices range from tantalizing cheap for the most useless knick-knacks, to prohibitively expensive for the most exuberant ones. Bartering for small items is guaranteed to invoke the shop owner's anger, but for more expensive items like jewelry, art, and carpets it might be worth a shot. Just remember, if you find yourself in a position of bartering, the shop owner across from you has tons of experience and will always come out on top.
Parallel to Pandrosou, Mitropoleos street houses most of the restaurants in the area and it too, will take you to Mitropoleos square. If you keep walking the straight line on Mitropoleos, eventually you will end up on Syntagma square, albeit after Mitropoleos street there is not much of interest, shops, or sightseeing.
Parallel to Mitropoleos, and diagonally to your left as you stand with your back to Monastiraki station, Ermou street would guide you to Syntagma square through a pedestrian, cobblestone pavement lined with the most fashionable shops. This was the main fashion district of Athens before the economy collapsed in 2009, but today you'll see more boarded empty shops as many open ones. If you walk Ermou street the other way (west), you will be walking through busy traffic and a more "gritty" sidewalk with fewer tourists. If you take that route, any left turn would take you right back into the pedestrian portion of Monastiraki.
As you orient yourself, with your back to the Monastiraki station, the impulse would be to walk to your left and through Ifaistou (Ηφαίστου) street. It's a cobblestone street lined with thrift shops and packed with pedestrians. While you can find tourist shops sprinkled here and there, most of the stores sell clothes, antiques, music, and home decorations. As you walk through Ifaistou, you will come across Agiou Filipou street. Going straight would take you through less touristy paths (and more gritty ones), but turning left will bring you to Adrianou.
Adrianou is a street that is lined with restaurants and cafeterias where you can enjoy a meal or a drink flanked by the ruins of ancient Agora. Walking to the west (to your left if you are coming from Agiou Filipou) would bring you full circle back to Monastiraki, while walking to the right would bring you to Thesion Metro station. From there, a left turn up the wide cobblestone street would bring you to the foot of the Acropolis. Beyond that, the pleasant walk would take you to the new Acropolis Museum and the Olympeion beyond.
Monastiraki shops are open all day long and they close at 10:30 PM (or 11 PM if there are enough people on the streets). Restaurants and cafés stay open though so you can enjoy a late meal or a drink. Best time to go is after 8:00 PM when the heat of the day begins dissipating, or in the morning before 10:00 AM.
Monastiraki is the perfect destination for your last day before you fly out of Greece. A dinner and a drink at an outdoor restaurant next to ancient ruins in full view of the Acropolis can be a memorable last day experience of Greece, and if you neglected to buy gifts for family and friends back home you can pack a suitcase full of knickknacks you buy in the myriad of tourist shops of Monastiraki.
Besides the overwhelming number of shops that sell tourist knickknacks, Monastiraki has it's fair share of quality shops that sell everything from original artwork, to hand-made craft items, to jewelry, to health food items. One can be lost among all the choices, and prices can vary from shop to shop, so do a round or two of window-shopping before you commit to an expensive purchase. Small souvenirs don't have such a big variety in prices.
While bazaar atmosphere makes it seem that it's a good place for haggling, shop owners frown upon the idea and might not be very cooperative. In terms of quality, the little souvenir items are the same you'll find everywhere in Greece (and every other country), gold jewelry, leather goods, carpets and fine art are of good to excellent quality. Monastiraki is also a good place to stock up on t-shirts, but be aware that imitation goods might be posing as brand name items in high prices.
You will also be amazed by the quantity of antique shops and thrift stores in the area. It is a well know spot for buying old vinyl records, WWII paraphernalia, and other assorted junk and treasures displayed in storefronts and open air kiosks for all styles and tastes. Speaking of taste, you will also be amazed at the amount of tasteless artifacts depicting sexual content (much in the form of ancient Greek erotic art) that are on open display.
If you buy electric items like the millions of night lights with Greek themes, remember that electricity in Greece is 220V and they might not work back home if your wall outlets only give your 110V. For larger items like carpets, shop owners have always had to answer the question "how do I take it on my flight?", and they have very creative ways of packaging your goods for you. For things that are impossible to squeeze down to a hand bag, most shops will ship your purchase to your home for an additional fee.