What is it called when you have a supple horse under you, hind end engaged and mouth in hand, and how do you do it? I always watch and hear about riders that talk about it.

Although I hate the term, it's called "on the bit". On the bit actually refers to the action of coming up from behind, using the horse's body, with the horse's power going up and forward through the hands and into the bit. As I've said on this site before, "on the bit" refers to the final symptom of the action (i.e head carriage) that will fall into place when everything else is working correctly. This issue with the term is that many people believe that on the bit is the first thing you do, and not the last. It's great that you recognise that this is not the case.

There is a German system of training that has been adopted on a global level and is now held in high regard by most riders. It's the Scales of Training. To achieve what you want to do, I highly recommend you research this. I should point out that it not a quick fix. This is the end goal for a lot of people's training, and can take months to years, depending on how far along you currently are in your and your horse's training. I replied to this post with a more thorough response on the Scales of Training: https://www.horsealot.com/answers/5a623b179a6d9cea198b4567/what-should-work-fist-when-starting-horse-dressage
You can just keep a light contact on your horse and they move nicely under you without you having to constantly give them the aids to do so. Every horse is different, with mine I open my chest and hips and drive her with my upper calf
Join the fun and sign up to connect with our 200,000 members!