One of the oldest cities in the world, inhabited for over 5000 years, Athens boasts so many stories that a lifetime wouldn't be enough to get familiar with them all. Perhaps the most notable is the story of Athena who the town is named after.
Athena, the ancient goddess of wisdom, competed with Poseidon for the honor of being a patron of the city. While Poseidon produced a spring by striking the ground with his trident, Athena created the olive tree. The townspeople accepted the olive tree as a more worthy gift and thus named the city after Athena. The olive tree is said to symbolize peace and prosperity. Even after it had been burned down during the Persian Wars, a shoot sprung from the stump, which indicated that Athena still her mark on the city.
The Parthenon
The area surrounding the Acropolis
The view of the Acropolis from Filopappos Hill (a public park with various paths leading to it - about 15 min walk from the Acropolis)
Athens at night
How much of Athens can you see in a couple days?
We had two and a half days to spend in Athens and that was just right. I would suggest scheduling at least half a day for full visit of the Acropolis. It takes about 30 minutes to walk up the hill and then it's nice to spend at least another 30 mins to an hour exploring the area of the Parthenon. We got up there just in time for the breathtaking sunset but that also meant we had less time to spend up there as they were closing.
Walking around town and exploring the Greek taverns and stores can be done in one afternoon as well depending on how much time one spends in the stores. Fillappos Hill and Lycabettus Hill both offer great views of the whole Acropolis. They are both hills - so there is a nice uphill walk (or a run in my case :-)) to each - but the views are well worth it! There is much more to do in Athens and in the surrounding area but if you happen to only have a day or two in between trips around Greece, that should be sufficient to see the essentials.