In order to unlock this achievement, you have to lasso one of these horses in the wild and break them. All three horses can be found in all three areas of the game, but some are more common in particular areas than others. The way to track whether you've broken them or not is to go to the local General Stores and look at the last three horse licenses available. At the start, they are all marked "unavailable". Once you've broken one of the horses, they will become available for purchase.
The Kentucky Saddler - A golden-colored horse with yellow/gold hair, this one can be broken during a quest/mission that you do at Bonnie's ranch. When you go out to break horses with her and her father, she will tell you to break a golden-colored horse. That is the Kentucky Saddler. thanks to geegers78 for pointing out that after this mission the deed will be available but you may still need to break another saddler.
Hungarian Half-Bred - A white/grey-colored horse with a black nose, this one can typically be found in a plain to the east of Escalera. If you look on your map in-game, you'll see this plain marked with three running horses on it. You can also find them in the far east of Mexico, out by El Matadero and Casa Madrugada. Also, as pointed out by Jack Burton, you have to break a Hungarian Half-bred for the stranger task: Love is the Opiate. thank you.
American Standardbred - A completely black horse, this one can typically be found in a plain to the south of Beecher's Hope in West Elizabeth and occasionally in Tall Trees.
Again, these are the MOST COMMON places to find these horses. It does not mean they are the ONLY places. Rarely, you might find one of these horses accidentally mixed into the fold in New Austin or West Elizabeth. Credit goes to Jakob187 for the clarity.
In the event that the achievement does not pop upon breaking the last horse it is suggested you first check for the three specific horse deeds at any general store. If they are all there then simply go out horse hunting and break them all again and it should pop as CHEEZUS points out. You can't just buy the deeds, you have to break the horses.