If you cannot log in to Unicount with your Estonian e-resident card or resident ID-card then most probably your computer is missing the latest version of Estonian ID-card software called DigiDoc or your browser does not have the necessary plugins installed and enabled.
Some possible solutions:
Check whether your card works on the government site eesti.ee
If it did not then check if your OS and browser support Estonian ID-cards
If they do please install the latest DigiDoc software on your computer
Make sure that you have the browser plugin installed and check that it is enabled
Then restart your computer
You can also follow this step-by-step guide or accessing web-based services such as Unicount.
Digidoc software and browser plugin installation manuals can be found HERE
If still no luck we would suggest trying to use your card to activate the Smart-ID signature app on your mobile device. It has the same legal status as ID-card in Estonia and with it, you can log in and sign without a card reader on any smart device.