You are going to have to be more specific. This sounds like a take-home essay prompt, and it would require an essay to answer it. Here's the cliff notes:

In the first phase of the war, the Spartans invaded Attica every Summer and destroyed everything in the countryside they could find. The Athenians remained behind their Long Walls (city and harbour at Piraeus connected) and let it happen, because they were rightly not dumb enough to face a Spartiate army on land in pitched battle. This meant that for almost ten years, the Athenians were not growing crops. The Spartans essentially controlled the land war (siege was not an option for a number of reasons I won't go in to).

But this didn't matter, at least not at first. The Athenians ruled the seas with their enormous and competent fleet of ships, and they were able to extract resources from their "imperial" holdings in the Aegean and Black Sea in order to feed themselves and keep paying for the ships (and everything else). They, in turn, harassed the outlying territories of the Spartans with little hit-and-run operations. The Spartans were, at first, very unsuccessful at meeting the Athenians on the seas -- their fleets were smaller, and did not have the experience. Ships are also very expensive to build and maintain, and neither Sparta nor her allies were particularly flushed with cash in the early war years.

Edit: if you would like to read about a momentous battle between the Athenians and Spartans, check out the Battle of Sphacteria in 425 BCE.

Long story short: over the course of the war, Sparta began to get batter on sea, and got some cash infusions (from the Persians, eventually). Eventually they started winning and slowly separating Athens from her overseas allies. Athens also made a series of serious blunders, including the disastrous invasion of Sicily in 415.

Could you please specify which military encounter you mean? They all had their own particular context and impact, and a post covering all of them would end up rather long.

• In 510-507 BC, the Spartans first helped the Athenians depose their tyrant Hippias, and then attempted unsuccessfully to install a pro-Spartan oligarchy in Athens.

• In 460-446 BC the Athenians and Spartans fought the First Peloponnesian War, ending in a thirty-year truce.

• In 431 BC the truce was broken and the Athenians and Spartans fought the Archidamian or Ten-Year War (431-421 BC), the first phase of the Peloponnesian War, which u/Alkibiades415 talks about in his post. This ended with the Peace of Nikias.

• In 414-404 BC the Athenians and Spartans fought the Ionian or Dekeleian War, the second phase of the Peloponnesian War. This ended with the defeat, surrender and dissolution of the Athenian Empire.

• In 395 BC Athens joined the alliance of Boiotia, Corinth and Argos against Sparta, and fought the Corinthian War (395-386 BC), which ended with the Peace of Antalkidas, also known as the King's Peace.

• In 378 BC, Athens joined the Thebans against Sparta in the Boiotian War (378-371 BC). The Athenians focused on rebuilding their naval supremacy and decisively crushed the Spartan attempt to challenge it.

In 369 BC, Athens forged an alliance with Sparta, and the Athenians fought on the Spartans' side against the Thebans for the rest of the decade. Since Sparta at this time had been reduced to a local power, Athens and Sparta were never again in direct conflict.