EAPI is a version defined in ebuilds and other package manager related files which informs the package manager about the file syntax and content. It is, in effect, the version of the Package Manager Specification (PMS) that the file adheres to.
The EAPI versions (and their meaning) are defined in the Package Manager Specification. Currently, EAPIs 0 through 6 are defined and officially supported.
- EAPI Usage and Description (Gentoo Development Guide)