You're thinking of Argos, and specifically the Battle of the 300 Champions at Thyreae, dated to around the middle of the 6th century BC. The event is reported in Herodotus 1.82.

You can read Herodotus's account for yourself, but to summarise, Herodotus said that at this engagement the Spartans and Argives agreed to match 300 of their men against each other in a fight to the death. At the end of the battle, the two remaining Argives returned home after they thought all the Spartans were dead, but in fact one Spartan, Othryades, remained alive and stayed at his post. The two sides each claimed victory, and a wider battle ensued, which the Spartans won. Despite the victory, Othryades, for his part, is supposed to have committed suicide out of shame for being the only one to survive.

The battle did in fact mark a historic turning point, since the two cities can't really be said to have been equals after this -- this marked the end of Argos's status as a serious contender for supremacy on the Peloponnese, and from this point on Sparta was always the more dominant city on the peninsula, controlling it through the alliance system that became known to modern historians as the Peloponnesian League.

Argos continued to be a rival to Sparta, though, and played a reasonably important role as an ally of Athens in the run-up to the Peloponnesian War in the late 5th century -- the Athenians' decision to ally with the Argives was an affront to Sparta that marked the breakdown of the Spartan-led system that had been established in Greece after they defeated the Persians. After Athens' defeat, though, Argos suffered further decline.

Some general sources: Donald Kagan, The Outbreak of the Peloponnesian War, Part 1, and Terry Buckley Aspects of Greek History 750–323BC, Chapter 4.