Each character has 6 stable slots by default. Based on your skills in animal taming, animal lore and veterinary you can have up to 20 stable slots.

100 Total Skills = 10 Slots
200 Total Skills = 14 Slots
250 Total Skills = 17 Slots
300 Total Skills = 20 Slots

Pets currently do not decay when stabled even if you do not refresh them.

Patch 53 Update: Each bonded pet slot you earn from the zookeepers quest expands your stable slots by one as well.