Baby pets are the ones from the quests. They cast heal on you. They quests are at levels 24-26 and take 10% of your experience while they are out. You cannot have another pet summoned while a baby pet is active.
The tamed pets are for level 60+. You go to the beast farm outside of ..Rune?.. and get special food (crystal or golden spice) then tame the critter. It follows you around for a bit (takes no exp from you) and gives helpful buffs as well as hp or mp regen depending on your class. It will randomly (as far as I know) leave you and may not teleport with you. You can have a pet or summons out with the tamed pet but you cannot control the tamed pet in any way.