Jewels are items that have mods just like your weapons and armour. However, instead of being equipped, they are socketed in the passive skill tree. Jewels only work when they're socketed in a jewel socket that has been allocated with a passive point. If they are in an unallocated socket, the jewel won't do anything.
Just like other items, jewels can be normal, magic, rare or unique. They can be modified with currencies like other items, though unlike your weapons and armour, a rare jewel can only have up to four mods. Some unique jewels, such as Energised Armour above, can modify the passive skills around them:
The colour of a jewel affects the likelihood of it having certain mods, but has no effect on your ability to socket it. Since the number of jewels you can socket is only limited by the number of jewel sockets you've allocated, some unique jewels have limits placed on them. If you have more of that type of jewel than the limit allows, the extra jewels won't have any effect.