API integration is a way to integrate outside services and applications into your own services and applications. It's a great way to save time and money for repetitive tasks on the web or to display information from another service in your website.
The most common example of an API is a shopping cart. The payments are handled by an API such as PayPal or Authori.net.
Some other examples would be a setting up Word Press to automatically send a tweet every time you add a new blog entry, or to automatically add purchases from the site to your bookkeeping program.