Stardust is a consumable resource that can be used to increase the CP level of your Pokémon by powering up. Stardust is acquired by capturing Pokémon or by putting one in a Gym and claiming the Defender Bonus. Note that Stardust will not help your Pokémon evolve into their next form, as that is Candy's role.
The amount of Stardust required to power up a Pokémon is proportional to the percentage of the Pokémon's maximum potential CP. There is no difference in the amount of Stardust required to power up a Pokémon's base form versus its evolved state.