The protagonist of the game is a boy named Clayton. Clayton's father has developed a formula to manipulate clay into animals. A witch doctor wants the formula and after Clayton's father refuses, he changes Clayton into a ball of clay and steals away both his father and the formula. Clayton vows to save his father and embarks on a journey through his backyard, the Pacific, Japan, Africa, and finally Outer Space. The player uses bits of formula lying around in these levels to transform into animals to best fit with the environment and make it to the end of the level. Special abilities that are in a clay ball transform Clayton into helpful animals as he races the clock to collect as much as possible and still make it to the end, jumping over obstacles and attacking with the animal in use. Clayton must also be careful not to be hit while a blob or he'll die.
When in the form of the Rat, one could witness the "Blaze-processing" touted on the game's box. This buzzword being a reference to Sega's claims of "Blast-processing" as a feature their Genesis console had over the Super NES. It wasn't often one could see a character in a Super NES title move as rapidly as Sega's Sonic.
Source: Wikipedia, "Claymates", available under the CC-BY-SA License.