I will try to explain it.
Euroleague is made up of the best clubs around Europe from all (or most) of the countries in Europe. It is the toughest competition in Europe.
All the clubs that participate in the Euroleague also play in their country's league and cup competitions. Fener just won the Euroleague by beating the Greek champions Olympiacos. Fener still has Turkish league play offs to worry about. I do believe that Turkish cup competition has already completed.
Essentially to qualify for the Euroleague, the team needs to be a champion of their country's league or one of the top teams that need to go through the qualification tournament. Obviously Euroleague champions of the previous year qualify directly for the Euroleague the following year.
There is a bit of misconception about the Euroleague in the USA. People think that is the only league the teams participate in which is not correct. Teams are still obliged to compete in their country's league plus the Euroleague if they have qualified for it. If they haven't qualified for the Euroleague then they play in their country's league and the Eurocup which is sort of a 2nd tier competition to Euroleague.
Clear as mud?