Trophies are currently only available on the PlayStation 4 version of Elite Dangerous. There are 58 trophies that can be earned.
Trophies require some online (open) play to get 100% trophy completion. Not all trophies can be acquired in solo play. Frontier has taken into consideration how much online play is required to achieve the Trophies. Platinum can be achieved without the need for the Horizons expansion.