The extended gaits are medium to upper level movements that are generally not taught to the horse until he has mastered lateral work, collection and his medium paces. Explaining the aids is difficult as they vary depending on the training and each specific horse. Basically, the extended trot is an energized trot where the horse takes the same height of movement as in the collected trot and elongates the stride to the point of maximum possible extension. The frame will also elongate to allow for this. The walk and canter are the same.
I think you are asking about a more basic movement, called a lengthened stride. This is performed in trot and canter on a basic level horse. The purpose is to show that the horse is relaxed and obedient enough to go more forward and open his stride (and come back) without going faster, either by cantering from the trot or quickening the tempo. The basic aids are simply to ask the horse to go more forward and elongate the frame by stretching and reaching. For me I go through the corner and half halt to balance the horse and once I am going straight on either the wall or diagonal, I use the extra balance and energy created from my half halt to lengthen the stride by pushing my hands forward and applying leg to encourage the forward, reaching motion. I make sure to refresh the balance and energy by applying half halts (read: re balancing seat) through out the lengthening. At the end we also need to be sure we are riding the horse back to the working paces, and not allowing him to simply fall back.