Essences is a group of currency that is used for crafting in Path of Exile. Every essence, when used on normal rarity item (white), will turn it into a rare item with 1 guaranteed mod.
For example, when you use Screaming Essence of Wrath on a white helmet, it will turn into rare helmet with 36% – 41% to lightning resistance (guaranteed essence mod) and several other random mods.
Essences are divided into tiers and 3 essences of the same tier can be upgraded to 1 essence one tier higher. I have attached visual reference for essences that I have found extremely useful, it offers an overview of all essences.
You can obtain essences by encountering monsters that are trapped in ice crystals. By clicking on the crystals several times, you release the monsters, the middle one carries the essences. After you defeat the monster, it drops all the essences it’s carrying. In the image below you can see how essence monsters look when you encounter them while playing.
In the image you can see Weeping Essence of Doubt – let me explain a bit: “Weeping” is prefix and it represents the tier of the essence. Other tiers you might see is wailing, screaming, whispering, muttering etc. “Of Doubt” is a suffix and it represents type of the essence – different types of essence offer different mods.
For example, Essence of Woe offers energy shield mods when crafting armour but Essence of Greed – life mods. To see all the mods and all the essences, check essence page on Path of Exile wiki.