If you think of a business process as a sequence of steps, then you can think of a process instance as a specific item going through those steps.
For example, in a business process which handles purchases, each instance will be an individual purchase order. There can be any number of instances traversing a business process, as there can be any number of purchase orders going through a purchase order system.
Every instance has a specific history and properties. A purchase order usually contains a customer name, a list of items, an amount due, dates of delivery and payment, and other required data. An instance will also have various status conditions. In the case of a purchase order, you would want to know if it has been approved, if it has been paid for, or if the requested products have been shipped.
Each instance has a beginning and end, as defined in the business process. As the instance proceeds through the process, it will be worked on by various participants or processed automatically by software.