This achievement is for riding/interacting with each of the beasts/animals.
Shock them with lightning to stun and then mount with the "Y" button, sometimes more than 1 shock is required to stun them.
All these animals are killed in hunting missions, so multi-task and do them at the same time.
The far left of the map has the Raptor elk, Razorfire Boar and Tollusk.
The iced area in the middle of the map has Hextadon.
The jungle contains the Laser Panthers.
And finally the Reaper Steads and Guillotar are further south of there.
You can ride the animals which you summon, but they don't count towards the achievement.