An electronic identification ("eID") is an electronic identification solution of citizens or organizations, for example in view to access benefits or services provided by government authorities, banks or other companies. Apart from online authentication many eICs also give users the option to sign electronic documents with a digital signature.
You may find hints for troubleshooting in the official documentation (Dutch) but note that Arch Linux is not officially supported.
Install theAUR, AUR and AUR packages, with dependencies AUR and AUR. These applications will automatically appear in your application menus. You can also start from command line with qdigidocclient and qesteidutil.
Native Messaging host for Google Chrome/Chromium and Firefox and it is the modern way of doing authentication and digital signatures on the web.AUR contains
For Google Chrome and Chromium you also will probably want to run esteid-update-nssdb script that enables TLS client authentication in the browser.
If you still cannot login from Firefox with your ID-card you should install AUR.
If you updated your certificates due to https://www.id.ee/?lang=en&id=38241, then for signing you need to use a patched version of opensc (see https://github.com/OpenSC/OpenSC/issues/1176). Consider AUR.
There once was an AUR package but it no longer works, since the
source isn't available anymore. 
The free implementation of BankID FriBID also no longer works with the current version of BankID.