Side reins are fixed reins that run from the bit to the front of a saddle or to a strap around the horse (surcingle). They're used when lunging horses to teach a horse to give to the bit.
Unfortunately, side reins don't have the feedback a rider does - and because of that a lot of trainers feel they do more harm than good, especially on young horses. They're often considered an "old fashioned" thing - in the bad sense.
Side reins are also sometimes used when riding, often to prevent a pony from pulling a kid over his head by the reins. (I personally prefer to use a very loosely fitted bearing or overcheck rein for this).