To equip an item card, talk to the Tamer NPC (again. what would we do without them') and choose to 'Creature Management'. A familiar looking creature control window will open. Drag your item card onto an empty slot for your creature and it's done.
You can do just the opposite as well, by equipping the creature card on yourself! You must rappelz rupees belt in order to do so. Just drag the creature card onto your belt slot to equip. Different stat bonus is given depending on the creature Card skill. The creature must not be formed in order for it to be equipped . If it already is, you must take it out of your control at the Tamer NPC before you can equip it. Because of that, the equipped card creatures do not gain any Exp or JP. You can equip only one creature card on your belt at a time.
Usually you would want a fully-grown creature card to equip on your belt for stat bonus, and if you are currently raising one that is different from your job class to aid you better (e.g. a melee class having a healing/buffing creature), you would have to buy another one you need. That's where trading comes in.
You can trade the creature cards with other players without any sort of penalty. Despite the rumors, there is no loyalty factor or anything like that. They will have their full stat and skills up to the level you have raised them when you give it to someone else.
V. Mountable Creature
Mountable creature, namely Ornitho , are not tameable, and can only be bought at Adventure Guide NPC. When you do the quest in Trainee Island, the Adventure Guide will give you one that lasts for an hour for free. Usually one costs 1000 rupees and lasts for 6 hours (not a bad price if you ask me). They're about 3-4 times faster than running even with extra movement speed, so it is handy to have one around when you're hunting in places far from any city.
In order to ride it, you must first summon it (if you have another creature summoned it will automatically be recalled). Select the creature and use Mount action. You must be near the creature in order to mount it, of course. If you are too far when you try to mount, your creature will simply move to the crature but not mount. Use the action again in that case.