Your web presence is how the public looks into your product, service or overall message. Your content however, isn't always managed in one spot. Maybe you manage events on a third party site such as Eventbrite, but you want your visitors to have as much of a seamless experience as possible.
API Integration is where it's at
Long story short, data that is held elsewhere is brought into your site so it can be displayed as native content, yet still managed separately. The depth of API integration is not only up to the developer building your site, it's up to the information and capabilities available on the third party that is serving up the connection.
Eventbrite for example, allows a web service to pull event information for an account. This information could be made into content on your site where you can change how and where it's displayed. When a visitor wants to register or for an event, only then would it take them to the Eventbrite site where all their information is managed.
Though the user still has to navigate elsewhere for some functionality, the process is quite a bit more refined than simply adding a widget to the page. Using the data gathering and maybe payment processing on the third party site takes quite a bit of load (budget and liability) off you.
Some APIs may allow complete user disconnection from their service while visiting your site, and some go un-noticed by the end user. Collecting e-mails and feeding them into a mailing list service, or looking up shipping rates directly from UPS or even displaying the current weather from various weather sources are all examples of commonly found integrations around the web.