Starting with the third game update Goo arrived in the house, causing "buddies" to become available by playing mini-games and obtaining specific scores. They are represented by gift box icons in the "Your High Scores" screen.
Fuzzball: Isn't he cute? - Something Fishy Level 2
Humming bird: Nature's helicopter. - Something Fishy Level 3
Butterfly: Does butter fly? - Shutterbug Level 2
Ptero: A Jurassic buddy. - Shutterbug Level 3
Bat: It runs on bat batteries. - Go Po Go Level 3
Dragong: Fire up your buddy! - Unknown
Vaporwish: It's made of thin air. - Unknown
Owly: Hoot if you think he's cute. - Secret Code
Vampy: He bites in small bits. - Go With the Flow Level 2
Ghost: He's hauntingly handsome. - Unknown
Witch: Pilot at Broom Airlines. - Go With the Flow Level 3
Numbskull: Beauty is on the inside. - Balloon Attack Level 2
Zombean: 50% dead, 100% fun! - Balloon Attack Level 3
Whirlybird: The sweetest chopper. - Jet Car Jump Level 1
Clyde the Cloud: (no description yet) - Jet Car Jump Level 2
Turk: Let's talk turkey. - All Washed Up Level 1
Sequoia: An expert pathfinder. - All Washed Up Level 2
Snowman: (no description yet) - Unknown
Rudolph: (no description yet) - Unknown
Elfo: (no descreiption yet) - Unknown
Flying Hearts: Love is in the air. - Unknown
Thanks to various people on the Never Forgotten Forums. The official thread for buddies is here, so feel free to consult that in case there have been updates I haven't added to this page yet.