The rest position in a wave is the position in which the wave would sit if there was no disturbance moving through it, which is sometimes also called the equilibrium position. The amplitude of a wave is measured as the distance from the crest of a wave to its equilibrium point, or rest position. The rest position can be thought of as the mean line through a wave.
In a transverse wave, particles are displaced and oscillate back and forth in a perpendicular direction to the direction that the energy is being transported, allowing energy to pass through them from one end of the wave to the other. This is in contrast to a longitudinal wave, in which particles are displaced in in a parallel direction to the direction that the energy is being transported. Transverse waves can be created and observed by stretching out a rope and "flicking" one end so that a wave moves through the entire rope to the other end.
The rest position of a transverse wave, which is also called the equilibrium, is the position that the wave sits in when there is not energy being transported through it. In the rope example, the rest position would be the position of the rope when it is stretched out and no waves are moving through it, which is a straight line from one end to the other.