The IRS taxes worldwide income of its citizens and green card holders.
Generally, for those Americans genuinely living/working overseas the IRS takes the somewhat reasonable position of being in "2nd place" tax-wise. That is, you are expected to pay taxes in the country you are living in, and these taxes can reduce the tax you would have owed in the USA. Unfortunately, all of this has to be documented and tax returns are still required every year.
Your European friends may find this quite surprising as I've heard, for instance, that France will not tax you if you go live and work in Germany.
A foreign company operating in a foreign country under foreign law is not typically required to give you a W-2, 1099, or any of the forms you are used to. Indeed, you should be paying taxes in the place where you live and work, which is probably somewhat different than the USA. Keep all these records as they may be useful for your USA taxes as well.
You are required to total up what you were paid in Euros and convert them to US$. This will go on the income section of a 1040.
You should be paying taxes in the EU country where you live. You can also total those up and convert to US$. This may be useful for a foreign tax credit.
If you are living in the EU long term, like over 330 days/year or you have your home and family there, then you might qualify for a very large exemption from your income for US tax purposes, called the Foreign Earned Income Exclusion. This is explained in IRS Publication 54. The purpose of this is primarily to avoid double
FBAR is a serious thing. In past years, the FBAR form went to a Financial Crimes unit in Detroit, not the regular IRS address. Also, getting an extension to file taxes does not extend the deadline for the FBAR.
Some rich people have paid multi-million dollar fines over FBAR and not paying taxes on foreign accounts. I've heard you can get a $10,000 FBAR penalty for inadvertent, non-willful violations so be sure to send those in and it goes up from there to $250k or half the value of the account, whichever is more. You also need to know about whether you need to do FATCA reporting with your 1040.
There are indeed, a lot of obnoxious things you need to know about that came into existence over the years and are still on the law books -- because of the perpetual 'arms race' between the government and would be cheaters, non-payers and their advisors.