In the context of the sale of a business, the "closing" is the point in time at which all necessary documents are signed by all the parties, apportionment of expenses up to the date of closing is done, money and keys are exchanged, and the buyer becomes the new owner of the business.
Sometimes the closing occurs on a particular day, when the parties meet with their respective lawyers to sign all the documents in each other's presence. In other situations, an escrow agent is used. Each party signs the necessary documents as they become available, and forwards them to the escrow agent over a period of days or weeks. When the agent has everything from both parties, he or she will release the funds to the seller and the deal is "closed." Whether an escrow agent or a closing ceremony is used really depends on the custom in your geographic locality, and the preference of your lawyer.
Immediately prior to closing, the buyer may want to do a "walk through" to be sure all the equipment and inventory listed in the purchase agreement is still there. Because accidents happen and inventory gets sold or broken, you should be prepared to do a bit of last-minute adjusting to the purchase price.