When your monsters are attacked, a blue bar will begin filling. Once it is full, you can tap the button that appears above your monster's head to have it use its special power. This must be done at the proper moment, before it gets killed. Set the game speed to normal before the blue meter fills in order to not miss the opportunity to use the special attack. Save fourty Gems to get the ability to summon a very strong monster.
Elements
Have a monster from each of the three elements (fire, water and nature) to increase your chances of having a strong attacker. An element is strong against one of the other elements, neutral against the same element, and weak against the remaining element. Water is stronger than Fire; Fire is stronger than Nature; and Nature is stronger than Water.
Upgrades
Each monster has a level cap. Use monster cards that are summoned with coins to strengthen your best monsters (at least four stars). Lower tier cards, even after leveled, will become outclassed later in the game. Try to keep your main cards even instead of maxing one out; you need at least one of each element at a decent level to defeat your opponents.