If the customer is a VAT-registered business in a different EU member state, the reverse-charge approach must be used: the recipient of the service is responsible for paying VAT to the local authorities. You must not charge VAT yourself, but must make a note on the invoice that the recipient is responsible per reverse-charge and must note both your and your customer's VAT number on the invoice.
If the customer is a consumer (anyone without a VAT number), VAT for certain services must be handled via the EU MOSS system (Mini One Stop Shop). MOSS makes it possible to sell digital goods in multiple EU countries without having to register with the tax authority in each country. But you still have to apply the customer's local VAT rate. If you have low annual turnover (below EUR 10k), you don't have to register with MOSS and can just apply your local VAT rates as if the consumer was in your country.