"Luck positively influences your trapping results. The higher your Luck, the bigger the bonus you get when catching difficult monsters." Your luck total will add to your difficulty levels when trapping, making monsters easier to catch.
As of the May 2013 update, Luck will no longer reset after the 5th consecutive day of play. Luck will continue to stay full as long as you continue to log into the game daily.
Your luck clover is displayed on the top of your game screen, next to your coins. As described by the in-game popup, it is easiest to hunt down difficult monsters when your luck is full. You must log into the game five days in a row to get full luck and your luck will remain full for every day you play thereafter. However, if you miss a day your luck will reset.
You can also get full "Uber Luck" by buying banknotes . The Uber Luck lasts for a week regardless of how often you play the game. You can click on your Luck Clover Icon to see how many days you have left of Uber Luck.