The business centre is an area with some desks, computers and a printer. Often there is a copier and fax device as well. You can access the internet, print documents, scan receipts and so on. The facilities vary a little bit from hotel to hotel. Often it is next to the hotel lobby, and not very private.
Usually the business centre is not very busy, because most business travellers will have a laptop.
If you go to a hotel without a dedicated business centre, the front desk will usually be willing to print documents for you.