I know that to be allowed to fly an aircraft has to be certified by an agency and that this one is not the same for European or American (for instance) aircraft. What are the different steps that an aircraft designed to fly in Europe has to go through in order to be certified? Proof on the paper of some features? Ground tests (which one)? Flight tests (which one)?
The EASA certification requirements are all publicly available online. Actually proving that an aircraft meets those criteria will certainly include all the steps you mentioned although exactly what happens probably depends a lot on the specific aircraft and manufacturer.
It does really depend on where the aircraft is being designed and manufactured. The initial type certification is done with the authorities in the State of Design / Manufacture. If that country happens to be outside of the EU, EASA will identify what extra proof of compliance will be required to fulfil the European certification requirements.
For the most part, the EASA guidelines (as issued, for example, in Certification Memoranda, and the FAA guidelines (as issued in Advisory Circulars and in guidance to those acting for the certification authority, such as 8110.105) are pretty much harmonized. (and that goes for the JAA as well.) There are some minor spots whre EASA is either ahead of or behind the FAA guidelines; For example, there is a paragraph of EASA CM SWE?? - 001 [don't remember the exact number] (dated 2014) that deals with a small class of LRU issues and that is not yet in any FAA guidelines that I could find.
Even though the fundanentals are closely aligned, you still have to demonstrate compliance to the FAA authority after having been certified under EASA, and vice-versa. However, since the process of obtaining certification pretty much demands attention to keeping your paperwork in order, your testing comprehensive and well-organized, and your certification artifacts available, once you have passed in one context the remaining work to be certified in the other is at least 90-95% in place.