New York is different than a lot of places. Your agent will come back to you with the counter offer or the acceptance of your offer. Then the seller's attorney will draft a contract of sale and will need to send it to your lawyer. You will need a lawyer. If you don't have one, I can refer you to ours who is fabulous - send me a PM and I'll forward you the info.
Before the contract is finalized you will need to get an engineers inspection - get some recommendations - there are a ton of mediocre engineers out there, and particularly given the age of the house, you want a GOOD engineer.
The contract will state the terms of sale, including any contingencies, etc. Once the contract is fully executed (signed by both parties), you basically will have to wait for the bank to approve your loan, and youre done.
Congrats!!! It's very exciting! and the house is adorable!