Teaching a horse to lay down is a particularly difficult trick to teach a horse. Remember that horses are fight or flight animals, to ask them to get off their feet and lay down, is asking them to give up the ability to either fight or flee. You must gain a horse's trust before ever asking or expecting your horse to lie down for you.
There are several methods to teach a horse to lie down- some involving ropes and actually forcing the horse down the first few times, but many horse trainers believe that method to teach a horse to lay down destroys trust rather than building it. For your horse to perform on command, in stressful situations like demos and clinics, the horse must perform the trick because they want to.
The problem in teaching a horse to lay down is associating the action with the command. All horses lie down, but because it isn't a general action when being worked with (or shouldn't be!) it's difficult to find a way to link action with the command. One way to do so, however, to teach your horse to lay down, is to find teachable moments. For example, if you know after a sweaty ride your horse will roll as soon as he's untacked and turned out into a certain type of footing- do it. Set the stage, stand at a safe distance, and as soon as the horse lays down, but before he rolls, praise him and repeat the command you'll want to use. (example: "Good Lay Down, Good Lay Down") If you train with a clicker, click at this moment. If your horse gets up right away that's okay, as long as the horse understands it was the laying down that earned the praise and/or treat.
Teaching a horse to lay down with this method takes a bit longer, but keeps handler safer and helps the horse perform more willingly long term. When teaching a horse to lay down, be very careful approaching the horse, if you choose to do so.