Catch rope? Is this like lassoing? I dunno. Anyway, here are the photos.
I am surprised, with all the horse work and breaking you have done, that you don't know what this is. It is the rope, (even visible in the photo) left on the halter, with a fair bit of length, which the horse drags around. This means you don't have to chase and frighten a horse that isn't used to being caught and handled.
Instead, you pick up the rope and as the rope is fixed to the halter, you can stop the horse running away, you turn the horses head to you and you can walk down the rope to the horses head, talking and soothing as you go.
This method is commonly used on horses that are shy or nervous, particularly on station horses or horses kept in large areas. Or where many horses are being broken or handled and the handler doesn't have time to spare chasing and cajoling.
It does pose a risk of course, with ropes getting tangled around legs, trees and fences and with horses freaking out. Usually though you find the horse learns to adjust to it. And even, as an added bonus, this rope tends to teach a horse to be more calm later in life when ropes (reins, halters, martingales, wire) tangle around legs.