Why Hungarian Horses?
Temperment - When people ask me what a Hungarian horse is like, I sometimes say, "Like a Thoroughbred on a good anti-anxiety drug."
When I was younger, I usually ended up riding horses that were either green or spoiled, because that was what I could afford, or what nobody else wanted to ride. I regarded it as a challenge. After breaking my back coming off a flaky horse when I was 21, and acquiring a kind, sensible (though green) horse when I was 24, I decided I really liked horses who were easy to deal with! Of course, every breed has variation within the breed - but of the Hungarian horses I have met so far, at least 80% have been exceptionally sensible, kind and willing. That has been one of the main goals of the breeders for over a century - and as I am now headed into my 50's, I appreciate temperment more than ever! My Hungarian mares are sensible, sensitive, trainable, kind and willing.
Gaits and Athletic Ability - Dressage, combined training, hunter, and driving horses 
The Hungarian horses have also been bred to have three good gaits, and to have a long, fluid stride that carries the rider comfortably. They have quite a bit of suspension in their trot and canter, so it is by no means like riding a gaited horse - but they are springy, not jarring, and their backs are quite soft and swinging. The traditional Hungarian Felver type horses do not tend to have the spectacular, flashy trot you see in some of the international caliber warmbloods (for that, look for a good Hungarian Sportlo, the cross between a Hungarian and a warmblood). The Hungarian horse was developed to have jumping ability, endurance, and agility - the officer's cavalry mount - the original three day event horse.
Durability and Endurance
A cavalry horse who breaks down or gets exhausted is useless - so soundness and endurance were the third focus of the Hungarian breeding program. These horses generally have better bone than Thoroughbreds, and excellent cardio-vascular systems. My farrier really likes the quality of hoof he is seeing in my horses. Although there are some quite large Hungarian horses, they tend to be more moderate-sized than warmbloods - and therefore longer-lived and less likely to develop lamenesses.
They are an endangered breed
The Hungarian horses are so few, and so scattered, that if we don't make an effort to continue breeding them, their wonderful qualities may be lost to future horsemen. Ironically, the fact that they cross wonderfully with warmbloods may be their undoing, as people breed more of the crosses, and don't produce enough of the Felver horses to maintain the original type.