When in Festival Plaza, you will be able to take on what are called Missions. These Missions range from answering quizzes to actually capturing Pokemon to searching for people who have answered certain questions in their profile, and you will receive Festival Coins based on how well you do.
When playing a Mission, a player with have the choice to host a mission or join one already in progress. To host a mission, a player must have a Festival Ticket. With these missions, players both locally and online can join when they are active. This can be missions being held by certain friends. When the mission timer is expired, your points are tallied, and other players that have joined will have their points added to the grand total. The higher the overall score, the more Festival Coins will be awarded to players.
In addition to these types of missions, there are also missions done through the Pokemon Global Link called Global Missions. These missions are ones that everyone playing these games can join online, and the result can mean a massive yield of Festival Coins if everyone succeed in this one goal. The goals set range in purpose, but the Pokemon Company usually sets date restrictions with these. To access Global Missions, a player will need to use Game Sync to the Global Link. After doing so, all they must do is Talk to the one character in the Festival Castle to check to see if there are any current Global Missions happening.