You click on Hunt a Critter and then click the pets that show on that page, there are 30 each time I believe. Finding the hunted creature is completely random, you may find it on the 10th click, or...
In the original Impossible Creatures game (without expansion packs), there are a total of 51 units, or creatures (though the main website claims to have 50 units), including:
33 land units
12...