It's called the EU Digital Covid Certificate. It can be digital or paper - it's a QR code. The official launch date is July 1, but people are signing up already.You'll need it if you plan to travel to or within the EU.
Currently, it's only available to EU citizens, but that's sure to open up.
No-one is forced to get one, unless you plan to travel, in shich case you'll also be carrying your ordinary passport too. Is this an invasion of privacy? Of course it is, and so are ordinary passports.
What is stored is: recent recovery from Covid-19, and negative test results -- including lateral flow, as well as PCR. It also records how many vaccinations you've had, and which vaccine.