Is there an API to retrieve airport departure and arrivals? I am looking for an API to get the arrivals and departures timetable for any airport in real time, like this table on the EuroAirport website: 



Are services like FlightAware or FlightStats relevant for me? I am a non-native English speaker so airport-specific language is a bit tough for me, I am not sure I fully understand what those APIs provide.

Edit : paying APIs are okay, I have a budget of around 100€ a month for this project
It appears that Flightaware does have an API (both SOAP and REST) that would satisfy your needs.  

Here is the documentation, see "arrived", "departed", "en route", and "scheduled".

Here's a code example from that page that gets data on flights en route to JFK and KSMO (this fits your arrivals criteria no?):



That said, you'll probably want to experiment with the API once you create an account and get a key to see if it meets all of your requirements.