There are ten, with corresponding strengths and weaknesses. I have just confirmed this in the beta:
Their strengths and weaknesses are as follows:
- Beast - takes 50% more damage from mechanical; takes 33% less damage from humanoid
- Critter - takes 50% more damage from beasts; takes 33% less damage from elemental
- Dragonkin - takes 50% more damage from humanoid; takes 33% less damage from flying
- Elemental - takes 50% more damage from aquatic; takes 33% less damage from mechanical
- Flying - takes 50% more damage from magic; takes 33% less damage from beasts
- Humanoid - takes 50% more damage from undead; takes 33% less damage from critter
- Magical - takes 50% more damage from dragonkin; takes 33% less damage from aquatic
- Mechanical - takes 50% more damage from elemental; takes 33% less damage from magical
- Undead - takes 50% more damage from critter; takes 33% less damage from dragonkin
- Aquatic - takes 50% more damage from flying; takes 33% less damage from undead
Individual battle pet skills have different postives/negatives in terms of doing damage to different pet types, so that isn't determined by the type themselves.