Besides playing league or friendly matches you also have the opportunity to participate in a cup competition.
There is a cup for each country. There are also some cups that requires you to win the best league in a country or get a certain vifa ranking.

You can sign up to a cup by going to World > Cups > Overview in the sidebar.

Here you can see the requirements you need to achieve to participate.
When you have found a cup you want to participate in, you can click the cup name and then the "Join" button, to the right.
If you want to leave a cup, you follow the same procedure, but click "Withdraw" instead.

You can participate in the cup that belongs to your own league. International cups are open to every club that fulfills the entry requirements.

If you change country during the season update you will need to join that country cup for the new upcoming season. You need to join the new cup before 22:59hrs that Sunday