From Monster Paradise Wiki
Your Stamina is a measure of how much you can quest. Each quest area has it's own toll for 10ft you are moving.
You will use ATK MP when you battle against Dark Monsters or other players for their Tablets. If your remaining ATK MP is too low, you will not be able to add any monsters to your party. ATK MP regenerates over time, or you can use a Battle Brew to fill back up immediately.
Your DEF MP is a measure of how strong a defensive party you can build to protect your tablets. Unlike ATK MP, this does not decrease when you are attacked.