To elaborate on what my trainer explained: cues from the reins are really the icing on the cake. In dressage the seat, body, and leg aids communicate 95% of all cues to the horse including gait, tempo, bend, direction changes, turning, etc etc. That is why I can ride my little green bean bridless--because of this concept. What my trainer means by headset is the soft "give" of the pole and also the very minor bend at the pole for circles etc. It really is not much, and it really is the last thing you think about- as in the last thing you ask for and the last cue you give. I have heard this described as riding a horse back to front.
Riding a horse on the bit has a definite feel. To me it feels like all things are possible and the horse can respond instantly to any cue for a transition or change of direction. Having a horse that is soft in your hands to me is the beginning of being on the bit. I think what I find confusing and what seemed to have others just as confused though is: is being on the bit the same as collection?
To me it is not. To me having a horse on the bit is just having a soft horse that is giving at the pole and when supported through the riders legs and seat encouraging him/her to be forward and reaching under your horse is now round and on the bit. I have always thought of collection being an advanced frame used in higher dressage movements like the collected trot, piaffe, and passage. I think on the bit riding is a definite precursor to collected work. Though as Kathy pointed out does a horse really need a specific head set to be collected? If you watch your horses playing and frolicking in the pasture I would say no (especially Kathy's horse Satin, I have seen her perform "airs" in the pasture that would make a Lipizzaner jealous).
That is why there is a dressage training scale. First you want forward. You want your horse to move freely forward no matter what. Then you want rhythm, and relaxation, balance and straightness. Only then do you want to harness that energy and start to shape it. Collection for me is a highly concentrated form of the horse's natural power and energy. When collected the horses front end will be light (so off the forehand) and the driving power will be coming from the behind. A collected horse is ready to achieve the amazing higher movements talked about above like the collected and extended trot, the piaffe and passage.

Recommended reading
Sustainable Dressage on collection
Jane Savoie on how to put a horse on the bit
Regarding horses on how to put a horse on the bit
By the by what do you guys think of the new lay out? Better? Worse?