The pet system (as it was in alpha) was pretty simple.
You couldn't buy a pet, however you get one at level 20 as a quest reward.
You also have a huge pet inventory.
There were 2 types of pets, trade and battle pets.
Each one boosted the appropriate stats.
Pets were obtained the following ways:
A low chance from farm boxes which you got through farming (Trade Class)
A low drop chance from all (or almost all) mobs (Battle Class)
GM Nyago's Pet Line (do it again Nyago!
As far as I know, pets were not captured, you only could get their eggs through the methods above, and then you use an incubator which you get from the general merchant and then incubate the egg.
Some eggs took 8 hours to hatch (8 hours of continous gameplay). The timer reset if you logged out.