In short: If you're supplying digital services to consumers, you need to charge them Value-Added Tax (VAT) in the country they're in, not the country you're in. If you're supplying to businesses, you charge them VAT where you are. You need to collect certain evidence with each transaction, to prove which category/country your customer is in.
The search term you probably want is "VAT MOSS". This officially refers to the "Mini One-Stop Shop" provided by HMRC (the British tax authorities), which tries to ameliorate the situation. You still need to charge people the right amount of tax, but then you can just upload a quarterly spreadsheet saying how much revenue you got from each EU country. But "VAT MOSS" is also used as a catch-all term for the whole cross-border digital VAT kerfuffle.
It started out as a well-intentioned attempt to close a tax loophole, but it was clearly drafted by people with no understanding of how computers work. (There are actually "FAQs" about how, precisely, to handle a German citizen with a British SIM card making a mobile transaction from a train journey crossing the Italian-Swiss border...as if this were something a website could reasonably detect.) But, as I say, there are decent solutions you can use, and you just spend a couple of days implementing one of them.