Edit: New information has been posted, check July 2014
Ok where to start...
Pets, they are your small fuzzy little friend. The is quite a large variety in which you can choose for a pet, however you can only have SMALL mobs as pets. It's quite obvious which ones you can have for pets and others you can not.
To start off, wait until you first reach Cemal. Roland will give you a side quest to find an adorable little mob as your tag-along buddy. He will give you a level 1 pet taming kit, but one will not be enough majority of the time. Go to the Mokar shop and buy a max of 9 more taming kits. (Find a map of the mob you wish to tame, save and do not switch maps. Begin taming and if you somehow fail ten times, you can go to the main menu without saving, and select your file again having ten more tries)
Once you have them, you can choose currently between a few mobs at the start of the story: Seed, Pod, Mush, Boar, Octo, Squee, Pitz (Or Sharp Ear Pitz), Corpi, Hornet, Crabby, Chikki, or Beeti. Each of these mobs are different in their own ways, and do not change upon taming. Ranged stay ranged, and close-combat is, well you know.
(The farther you are in the story the more mobs you can tame, but only the small ones. Later when you find the Hatchlings which are tiny element breathing dragons. Those are the best to have due to their high defensive nature and ranged attack with an element, however your pet being rare is kinda a higher value than them since they are VERY HARD TO TAME and from that the Rare Hatchlings are quite difficult to obtain. It be best to use a Special Taming Kit from the Cash Shop on them.)
Now upon taming each pet will be assigned 1 of the three types of pets. There are:
Helps fight monsters (Only becomes aggressive when other mobs become aggressive against you, will not attack bosses)
Provides those buff runes socketed on separate equipment, when equipped to the pet, will give you and your buddy the bonus. (Ex. Pitz with a 'rune of annihilation general scimitar' [+40 DMG], will give you both the DMG bonus)
Upon any monsters death, your pet will swiftly, and quickly harvest the loot. I've rarely found loot pets useful, unless your mobbing. (Have an entire map attack you)
Pros and Cons about each type:
Pro; Helps you fight, often being able to take/deal higher damage than yourself. !! Will not attack bosses !!
Con; Pet is vulnerable to any attack, and more often than not, will die fairly quickly from not nearly enough DEF, or DEX as yourself. (Coming from a Scout class point of view)
Pro; Any socketed rune equipped on your pets weapon/armor will be given to you and your pet.
Con; Pet will NOT fight, still vulnerable to any attacks, unless Rare or Heroic with combat as another option.
Pro; Will move directly towards fallen enemy, then often moves out of way if combat persist.
Con; Will not aid in combat, also vulnerable to attacks. Best to be paired with another option. (Rare/Heroic)
With that said you're probably wonder why I stated pets can be Rare/Heroic. Your first pet is most likely a General. Unless you go back and tame Puppy (Do not tame it, I will post soon on why) When you tame a monster, it will drop 5 levels of Exp. Then once it catches up in levels to be on par, there is a chance of your pet becoming Rare. You will get a notification on the screen about the success or failure. If your pet succeeds becoming Rare, at level 60 there will be another roll to see if your pet evolves into a Heroic. This roll is very very small for success. So try not to have your fingers crossed.
However there is an easier way to have a Heroic pet. The special taming kit from the in-game cash shop from AllStar. Upon taming your pet will become Rare automatically, and at level 60 (Same as the normal way) it will roll to succeed or fail in Heroic evolution. This roll however is bigger than the normal method. A better chance for your pet to have all three types, only trade off is you're paying real money for some data you'll toss away later.