There are two of them.
Spirit Island: This one is south of that one island in the southwestern sea, and in between 3 rocks. You know those gem thingies? Bring them to this pond and either your Power (Attack), Wisdom (Defence) or Courage (Speed) will level up, depending on the kind of gems you have.
Uncharted Island: When you come here, you will from that point on be able to teleport (when on the sea, of course). However, you must first solve a riddle. Hint: The map can help you out. What map, you ask? The map you drew while exploring!
Chat with our AI personalities
To Put A Fish in the Pond: You hold a fish standing next to the fish pond, then press A to drop the fish in the pond To Take a Fish out of the Pond: Stand in front of the brown sign at the pond. Press A and it will ask you what fish you want. Select the fish and you will be holding it.
You must have 10 of the same species before you could stock them in a pond.
If you go to Mt. Moon, you will eventually exit on a place where there will be a small pond. If you go there at night, supposedly there will be a group of clefairy that encircle the pond. Talk to the people living in the house near that pond for more information.
Unfortunatly, they don't have the koi pond anymore... it may, if you're lucky come back though. Good Luck!
The old man is no one of real significance to the game. He'll just tell you that he's seen Grimer in the pond.