Sometimes, unfortunately, the date of buyer possession was never clear in the first place.
This is when each of the parties, buyers, and sellers, may develop a contradictory expectation of possession.
In some parts of the country, buyers give the sellers a day or two after closing to move. Whatever you agree to, make sure it's in writing.
Your real estate agent will likely have a standard purchase offer or purchase agreement form for your use.
In some situations, or if you live in a state that requires attorney involvement in real estate transactions (as is the case in Alabama, New York, and North Carolina, for instance), you might want to hire a real estate lawyer to prepare or review the contract.
In other areas, you are hire an escrow or settlement company to act as an independent third party.
Make sure that there are sufficient funds in your account to cover your deposit check, as it will be cashed.