How do you get a second pet as the night elf in world of warcraft

If you are a Hunter, (it doesn't matter what race) you must reach a higher level in order to get more pets. You will begin your game with one LV1 Pet, and if you don't want it you can learn Tame Beast when you get to LV10 then abandon it and tame a new one. If you want more pets besides the one you have you level up for them. You'll need the spell "Tame Beast" learned at LV10 before you can tame more pets. You must find a beast you want and use the spell on it whilst your current pet is dismissed.
 LevelingCall Pet 1 - Known at Lvl1
Call Pet 2 - Learned at Lvl18
Call Pet 3 - Learned at Lvl42
Call Pet 4 - Learned at Lvl62
Call Pet 5 - Learned at Lvl80

Once you have the required level go to a Hunter Trainer and buy the spell. Now you have more spaces for more pets. You can have up to five pets. Tame Beast, Feed Pet, Control Pet, Dismiss Pet and Serpent Sting can all be learned at Level 10. Serpent Sting is not required for managing your selection of pets, but to get new pets you must use Tame Beast, to keep your pets at 100% Happiness / 125% Damage you must feed them, and to command your pet to use it's different spells and attacks you need Control Pet. Some other spells are learned at this stage. Before taming your new pets be sure which category of pet you want - Ferocious, which can kill targets easily but is easier to kill, Defensive pets, which can occupy enemies with their big amounts of health points, and then you have the Balanced category - they are (DUH) balanced.