System Integration is when you tie or establish communication between two or more computer systems. Say you have a CRM system that you want to tie to your financial system, you need a special set of APIs (application programmers interface) to transfer data back and forth between the two systems. More and more systems these days are offering EDI (electronic data interchange) capabilities that makes it easier to share data between systems so it is more convenient for you and your customers to get benefit.
An example will be say you have a database of your retail products with images, SKUs, Description, Specifications and Prices. You want to upload these product information to Amazon.com or eBay. You will use a system called "Middle Tier" app to customize/automate the upload process without having to manually enter them. Systems Integration are highly useful for getting information flow done seamlessly to save time as well as increase accuracy of delivery.