Blood Shards are an alternate currency in the Adventure Mode of Diablo III: Reaper of Souls. They drop from Horadric Caches and Rift Guardians. They can only be used to buy items from Kadala, through gambling. They can be picked up by simply walking over them, just like gold.
Rift Guardians on Normal through Torment I will drop the same amount of Blood Shards, but the amount will increase with every difficulty above Torment I. In Greater Rifts, the amount also increases with rank, and is generally higher than in normal Rifts on comparable difficulty. Greater Rifts are therefore a more reliable method of farming Blood Shards (at cost of foregoing normal item drops).
A player can possess a maximum of 500 Blood Shards at any one time (excessive amount cannot be picked up). As of patch 2.2.0, each rank of Greater Rifts beaten successfully in solo mode increases this limit by 10, stacking infinitely, for all characters of the same type on the account.