Power Berries are a measurement of your energy level. The more Power Berries you've discovered, the more work you can do before you exhaust yourself.
Next to your farm house is the Secret Garden. Each flower you see in the garden represents a Power Berry that you've found. There's no way to actually enter the Secret Garden since it is surrounded by stones. You don't have to water the flowers either. There are 10 Power Berries that you can find so you'll end up with 10 flowers in the Secret Garden.
Power Berry #1 through #6 - You can get these easily by just tilling soil on your farm using the Hoe. Power Berry #7 - A gift from Marla the Mermaid.
After Billy and Elza give you the boat, a note inside a bottle will appear on the beach in the Village. The note is from a lonely mermaid, who is looking for a friend. She will send 3 notes, about a week apart. You have to answer her notes correctly in order to earn the Power Berry.
Bottle answer 1 - "I believe you." Bottle answer 2 - "Be my friend."
After you answer her two questions a third bottle will appear on the beach. Marla is anxious to meet you and gives you instructions on how to get to her home.
Left, Left, Up, Up, Up, Right
Hop into your boat and select "No" to going to the Mainland, then select "Yes" to go sailing. Once you're in the ocean go down 1 screen first before you follow her instructions. Eventually you'll reach her sea cave. Go inside and walk up to the edge of her pond and press the A button. Marla will swim towards you and reward your kindness with a Power Berry.
Power Berry #8 - Failing miserably while playing the Snowboarding Game.
If you buy the Snowboard from Lucus in the Winter season you can use it in the Grassland. There is a secret Power Berry hidden in the downhill course that you go through. To find the berry you have to run into as many logs and trees as you can while you snowboard. You'll need to run into at least 20 logs and trees. When you finally reach the finish line (which should take a long time because you keep falling down!), you'll earn a rank of D. Your prize for doing such a horrible job will be a Power Berry. You don't have to earn a rank of S before the berry will appear. More information about how to use the Snowboard can be found in the Island Events section.
Power Berry #9 - Help Cozy fill the tanks in the Aquarium
At the beginning of your second year, go talk with Cozy at the Aquarium. He still doesn't believe the fish you have caught for him are drawing a large enough audience. Cozy says that the Elder has talked about a very rare fish called the King of Lake. You have 2 days to catch the King in the Forest. If you don't catch him in that time he'll never reappear and you won't get the berry. When you do catch the King of Lake, give him to Cozy. If that's the last fish he needs he'll reward you with Power Berry #9.
Power Berry #10 - Accept the bribe from the Harvest Sprite
On the 9th of Spring in the second year an earthquake will occur. The Island's volcano has erupted and made a mess of the Mountain. Go visit and you'll find a cave up the path. Inside the cave you'll find the Nevermelt Ice (for Lucus' Fridge conversion) and a secret path. If you follow the path (see screen shot) you'll find a sprite at the end. He's been hiding out in the cave and doesn't want you to tell anyone where he is! He'll give you the last Power Berry if you keep his location a secret.