For some people its a horse that every little move, touch, shift of weight or inhale/exhale of breath you make is a different cue. Your leg touches his side here and you get reaction X, slide back an inch and its a different reaction. And that's great IF you're a rider with that much finesse.
Another broke is a horse that excels at its particular job, even if it may not be too great at a something else. Like a barrel horse that is NFR quality but you wouldn't want to take over a course of jumps, a cutter that is tough on cattle but you wouldn't try dressage with or a reiner that you wouldn't take out on the trails (NOT that any of those jobs are exclusive to doing anything else, just that sometimes a highly trained horse in one discipline isnt' really good at much else due to the focus & time training the trainer puts on their discipline)
Then there's the horse that you can toss any greenhorn or dude on, any city slicker or visitor that comes out. He's the horse that may not be 'fancy' or showy or have lots of 'buttons', but he doesn't booger at the plastic bag blowing in the breeze. He may not win in the showpen but he's the one you trust with your non-riding parents, your friend from work that wants to 'go ride a horsey' or your nieces and nephews that don't really ride but have outgrown the pony. He may neckrein or plow rein, go forward, backwards and stop but that's about it.
All those horses have their place, and they all have value... sometimes with papers and sometimes without. You just have to know what your needs/expectations are.
to be continued...