There are five different companions you can unlock during the game. For some you need the "Bait Box" from Alex and different bait to tame them. Place the "Bait Box" near them and put in the bait. Go away a few steps and wait for the animal to near the "Bait Box". Then go to the animal and tame it.
Dog[]
Bait: none
Achieved: Reward for ending the Dungeon on starting island
Behavior: Fetches items for you
Cat[]
Bait: Raw Sardine
Achieved: Running around in Bridgewater on starting island
Behavior: Fetches items for you
Wolf[]
Bait: Raw Venison
Location: Appears only on "Harvest Moon"; main dark forest island
Behavior: Attacks enemies for you
Tiger[]
Bait: Raw Rainbow Trout
Location: Can be found on the desert islands in various locations.
Behavior: Attacks enemies for you
Red Panda[]
Bait: Bamboo
Location: cherry blossom islands
Behavior: Fetches items for you