NPAPI's lineage harkens back to Netscape 2.0 where it was used for extending the browser to support new content-types. Other browsers of that era implemented the API as well so it became a bit of pseudo-standard. It's been updated over the years in various ways, some bad, some good — allowing bidirectional interaction with JS and not requiring the plugin to render something to the page are probably the main improvements to note.

NPAPI is useful in the context of Chrome extensions as it is the only method (currently) available for running native code in an extension. Firefox extensions are able to bundle NPAPI plugins in this manner as well however this is a less common approach as XPCOM components are generally a better fit for that environment.

One notable difference between extensions in Firefox and in Chrome is that Chrome allows you to specify that a plugin is only executable by your extension; a handy thing as you may not want any arbitrary web page to run your plugin (XPCOM components are not by default accessible via any old webpage).

Internet Explorer no longer supports NPAPI which is why ActiveX support is a useful target.

Native Client as far as I understand is targeted more towards producing one-off executables rather than reusable components.