Your creatures can select from a decent variety of actions during battle. This page will explain how to utilize them. Upon selection of an action, your creature will call out the name in a bubble near it to verify the action that has been chosen.
Click this button to initiate a battle turn after all of your creatures have selected their actions.
Click this above button to repeat your creature's actions from a previous turn or battle.
Use these buttons within the battle log to control the speed and presence of battle animations.
Toward the top of the battle screen you can see the order creatures will execute their actions.
Turn order is calculated based on the amount of SPD and WHT each creature has. More WHT will make your creature less likely to take action before others, whereas more SPD does just the opposite.
You can hover over the images located in the Turn Order section of the battle screen for more details.
The turns allowed in a battle can vary depending on the type it is. This is to make it more difficult for the player to while it is "safe" during fights. Typically boss fights will have more turns than normal fights, but please be aware that this is not always the case.