API is an Application Programming Interface. It's not a 'thing' as such.
It's just a way for one lot of code to integrate with another. Every Joomla component uses the Joomla API, which is the collection of constants, classes, objects, global variables and functions that Joomla provides specifically for those components. This is just the same as how every Windows program uses the functions that the Windows API provides to interact with the user.
Hopefully that makes sense!