Congratulations! You just found out you're moving to New York City and are ready to start looking for apartments. Although you can prepare in advance for the search, you won't be able to actually sign an apartment lease until about a month before your move-in date.
The New York City real estate market is extremely efficient - at any given time there might only be a 1% vacancy rate. And since most landlords require a 30-day advance notice if you're not planning to renew your current lease, they may not know if an apartment is going to be available until just a few weeks before it becomes free.
That combination of very few free apartments and very short notice means that the vast majority of apartments that are being shown are for an immediate turnaround. Generally apartments will be available for either immediate move-in or move-in within a month of when you are looking. There may be a few exceptions if the landlord was given advance notice, but if you start looking too far out you will likely find yourself falling in love with apartments that would require moving in a month earlier than your planned date.
Even though you might not be able to rent your apartment more than one month out, it still may be worth looking at a few places before then. By taking a look a bit earlier, you'll be able to get a great sense for what apartments are available in your price range. That way when you you're playing for "keeps" in the month before your move-in date, you will already know a great apartment when you see it and can sign the lease without any second-guessing.
To recap, if you're looking to move-in, for example, September 1, a good time to begin your apartment search in earnest would be the first week of August.