Pets are monsters that are useful to the player. They travel with the player, helping out in battles and problems. You can catch them or buy them. Only members can catch all 100+ pets or buy Fishbol (Sold from the Merchant in Clockwork Town). However, Clockwork town is deleted.
After finishing the first 2 given challenges, you will be asked to chose one of 4 different pets. Each pet is a different element: forest, fire, water, storm,and ice. They all have the move in their element that does 1 heart of damage (mudball, fireball, water blast,cloudshot, ice cannon. However, there are no damage hearts anymore, and there are no ice pet starters.
Battling is what most people use pets for. They do damage to other players/enemies. Perhaps you are knocked out in a battle. You could easily switch to a pet, and they will avenge you. They do damage to other players' pets, depending on the type. Earth beats Water, Water beats Fire, Fire beats Earth and Ice, Ice beats Sky, Sky beats Water.
For more info, feel free to check out the Elements link.
Buying pets is one way to obtain them. There are many Merchants across the map, that sell different type pets, depending on the area. The Merchant at Barnacle Cove sells Squibble and Funkeel. The Merchant at at Clockwork sells the only Membership pet that you can buy, Fishbol (Clockwork town was deleted!!!). The merchant in Bonfire Spire sells an Ashlet and the merchant in Firefly Forest sells a Mystile. Finally, the merchant in The Skywatch sells a Nebluff and a Mystyyk, and the merchant in Shiverchill Mountain sells an Arcticlaw. All pets are 1500 in gold coins and are at level one when you buy them.
Catching pets is another way. Although some can be bought, most pets can only be caught. There are many out there to catch, too. You just need to go into a battle with one and click the "Catch" button. Once you do, it will charge you 500 gold coins and show a spinning arrow. When the arrow is in the green "Catch" section, click the catch button. Do not click the button when the arrow is on the red, or the pet will run away. Simply click in one of the empty spaces to catch the pet. Pets cannot catch other pets, so they don't have that "Catch" option. Only your wizard can catch pets, which is quite logical.