Resilience is an attribute that reduces the chance of receive critical strikes from either spell or melee strikes, it reduces the effect of mana drain spells and reduces the damage taken from DoT (Damage over Time) effects.
The resilience stat was brought out with the Burning Crusade expansion as part of the new combat rating system. Before the expansion, there was no such mechanic and, because of this, this rating only shows on items available to players over level 60, with some rare exceptions.
A player doesn't start with resilience, it is only acquired by things like equipment, enchantments and gems. The majority of higher end PvP rewards all grant players resilience.