The so-called "closing" is the final transfer of the house from the seller to the buyer. It occurs after both sides have met all the terms of the contract and the deed has been recorded. Closing also refers to the time when the transfer will occur, such as "the closing on my house will happen on January 27 at 10:00 a.m."
The event referred to as the closing frequently takes place at the office of the professional who handles the transaction -- such as a title officer or real estate lawyer. However, the buyer and seller don't always have to be there at the same time. It usually involves a lot of document signing.
You might expect that the closing would be the date on which you'd pay the seller all that's owed on the down payment -- and indeed that's the deadline for doing so. As a practical matter, however, your escrow or title company will probably recommend that you take care of all the money transfers a couple of days in advance, to make sure all goes smoothly.