It depends on what time period you're talking about, but for the most part, yes. During the era of "Classical Greece" Athens and Sparta were the two most powerful city-states. The Peloponnesian War (431- 404 BC) was fought between Sparta and Athens as each one built up its own "league" of allies/clients, The Athenian Delian League and Spartan Peloponnesian League and tried to gain hegemony over all of Greece. Eventually, the Spartans would emerge victorious, and Laconians were able to wield disproportionate political and military power as governors and administrators in other Greek cities. Thebes also deserves an honorable mention, though, and does not get enough credit in the general consciousness. Thebes was able to gain the upper hand at points during the military wranglings of different alliances of city states and was itself the most powerful city state at times, notably under Epaminondas.

Of course, these cities would eventually be overshadowed/conquered by Phillip II of Macedon, and other cities in Greece like Corinth or Thessalonica would grow in importance during the Hellenistic and Roman periods.

Beyond all of that, Athenian thought was incredibly influential through men like Socrates and Plato and Athens would remain a center for philosophy and education for centuries. Attic Greek also became the more educated and prestigious dialect of Greek, being the language Alexander was taught in, for example.

Hopefully someone else with more expertise can provide more details.

Slightly OT, but as a laymen, where can I learn about the daily lives of spartans? The structure of their society fascinates me and I really wonder what their day-to-day lives were like.

Has no one mentioned Mycenae yet?

Mycenae was the dominant power in a large part of Greece for a time, though this was centuries before the Peloponnesian War, and perhaps a bit before the real heyday of classical Greece kicked off. The whole time period is in fact called the Mycenaean Period, this was by the end of the Bronze Age. About 1600-1000 BC according to Wikipedia.

The Mycenaean Period was the time of the Trojan War by the way, if that actually took place.

Michael Grant's "The Rise of the Greeks" is a good read if this question irks you. Different cities were more or less influential at different times and in different ways. That's a serious hand-wave of an answer, but it's basically true. Some examples might illuminate that vague statement. Mycenae had a considerable hegemony in the earliest times, and later faded. Mycenae was part of the Argolid, and if we believe (with a dose of salt) Homer and the most optimistic of Hittite scholars who assert that the references to Ahhiyawa and Wilusa refer to the Trojan war, it seems that the cities of the Argolid continued to exercise influence for some time. In any event, Argos, a city near to and closely involved with Mycenae, vied with Sparta for control over the Peloponnesus.

Corinth was, at least at one time, the richest Greek city, with considerable naval power. You may also want to count Syracuse, which was a rich and powerful Greek colony. Athens and Sparta obviously feature prominently because of the War immortalized by Thucydides. Indeed, they remained relatively powerful as the rest of Greece was subsumed by other empires -- Sparta maintained control over its principal area as the rest of Greece was conquered by Philip II of Macedon, and the Roman Empire allowed Athens a great deal of self-control after the Macedonian wars, although the latter may be more of a cultural and intellectual deference than having anything to do with power.

I think most people have Sparta and Athens first in their minds when this question arises because they were the "last men standing" in a certain sense, and also because of the scale of the Peloponnesian War and the weight of Thucydides and his account of it.