It's a complex topic cause... it depends. I know a guy that lives in Germany and works in Luxembourg (he is the coffee delivery guy for a lot of people xD). If I remember correctly, he pays taxes in Luxembourg and has to fill out a form each year for the German tax office that he or more exactly his company is paying his taxes in Luxembourg. Cause one big thing in all the european countries I know is, that as an employee your workplace pays your salary based taxes. Most social contributions like pension payment is also a Luxembourg thing, but health care is a mixture. He pays the biggest part in Luxembourg but has to do a little extra payment in Germany, so he can visit a German doctor like someone that pays his healthcare only in Germany.

But it's different for every country.

And also another funny thing I know: The US is the only country in the world where you have to pay taxes just cause you're a citizen. No matter if you're living and/or working in the US. So for example a US citizen working and living in Germany still has to pay a (small) amount of US taxes. For example if I would work and live in Italy I would only pay taxes in Italy even though I have a non Italian passport.