Each island has different game on it but for the most part:

Axis Deer
Black-tailed Deer
Mouflon Sheep
Feral Sheep
Mouflon-feral Hybreds
Feral Goats
Feral Pigs
Feral Cattle
Turkey
Ringneck Pheasant
Blue Pheasant
Kalij Pheasant
California Quail
Japanese Quail
Gambels Quail
Bobwhite quail in some places.
Mourning dove
Laceneck dove
Zebra dove
Erkels Franklin
Black Franklin
Gray Franlkin

The Island of Ni'ihau has a bunch of african game on it. There's some other animals on various ranches in the state (I've shot bison). Pronghorn were released on Molokai in hopes that a herd would develope but until recently, the herd was kept low by poaching and might be gone now.