Application integration services can take on many forms, but a common goal is getting multiple systems to share data. Often the goal is to reduce duplicate entry or automate data flow between different systems. The systems could be internal, external, web or non-web.
Perhaps it's sharing customers from an accounting system with a customer relationship system. It is could also be integrating warehouse management systems with a web portal or integrating an external system to process orders.
Often, integrating systems allows companies to pick best of breed software instead of settling for an all-inclusive solution with fewer capabilites.