A BusinessProcess is some reproduceable process within an organization. Often it is a something that you want to setup once and reuse over and over again.
Companies spend a lot of time and money identifying BusinessProcesses, designing the software that captures a Business Process and then testing and documenting these processes.
One example of a Business Process is "Take an order on my web site". It might involve a customer, items from a catalog and a credit card. Each of these things is represented by business objects and together they represent a Business Process.
If you are familiar with ObjectOrientedProgramming a Business Process is usually encapsulated within an ObjectClass?. The goal of EnterpriseApplicationIntegration is to share both Business Processes as well as data. - DanMcCreary
I'm not so sure about the "reproduceable", "set up once", bit. As far as I can tell, a business process is a behavioural pattern within an organisation that helps satisfy stereotypical goals, and the goals and business process itself are continually adapted by negotiated goal alignment between interacting parties. So, for example, if a customer goes into a repair shop with a broken widget, the repair shop will have seen things like this before (i.e. a repair - a sterotypical goal) and therefore appeal to their experience of what has worked before (behavioural pattern) to work out how to deal with the repair. However, since they are dealing with a real live human being here, the humans at the repair shop will generally be using creativity and communication to negotiate a price, to agree a completion date, to handle any complaints about quality, possibly to perform the actual repair itself, and so on - in essence to align their goals with the customer's goals and, thus, hopefully strengthen their business relationship for future interactions. -- AnthonyLauder
See BusinessProcessExecutionLanguage for a WebServices way to implement BusinessRequirements.