To apply for a partner university you should research which universities you have the possibility to do your exchange at. When you have gathered this information you log into MOA. Under "University" you can select the universities you would like to study at and put them in the order you prefer. You can change your choices of universities and preference order until the application deadline.
Once the application deadline has passed it will be 1-2 weeks wait before you get the result of your application. When you get offered a place you can accept it in MOA and move on to getting courses approved.