Instructions
Switch to an Active Pet
Dismiss your current pet by clicking the red "Dismiss Pet" icon in its spell slot in the Action Bar at the bottom of your screen, in your spellbook or by double-clicking the mouse cursor on your pet.
Execute the "Call Pet" command in the same way that you dismissed the old one.
Click on the portrait of the new pet you wish to summon.
Switch to a Stored Pet
Open your local map and tag the nearest Stable Master's location.
Travel to the Stable Master's highlighted location and click on them to converse the same way that you would with a merchant or quest-giving NPC.
Click and drag the desired pet from the lower row of "stored" pets to a slot on the upper row of "Active" pets in the window that appears. This will bring the pet out of storage and into your inventory to be ready to be called. If you have no available slots, you will need to either abandon a pet or store one to make room.
Click the "X" to close the window.
Click the "Call Pet" spell and select your newly activated pet (See Reference 2).