Pets have essentially turned into WoW's equivalent to Pokemon, in the sense that you can catch wild pets, level your current pets, and battle other "trainers" for achievements, titles and what not.
Most pets can be caged and traded, with a few exceptions like BlizzCon and Collector's Edition pets.
These guides might help, they are extremely in-depth and detailed on the subject:
This post was edited by Arkhen on Sep 26 2017 11:44am