Tempers increase your HP, defense, and damage and can be obtained through PVP or PVE.
PVP Tempers work by participating in Arena fights and earning tokens. At the end of the season, you can use those tokens to buy PVP tempers. This is done by clicking on equipped gear and choosing an available temper level.
Buying a temper on a piece of gear that already has a temper, replaces the temper, losing the previous one.
PVE tempers come in three flavors: Defense, Physical, and Magical. Essences for each drops in typical combat with monsters.
Each type of temper requires a particular essence and a single silk of a tier equal to 1/3 of your xp in billions (up to 90b) OR a rune of any type. Example: if you have a total xp of 49 billion, you will need a t17 silk.
To use PvE tempers, you need to go to loc 9,29. Then use the enchanting rod with an equal stack of PvE tempers in the top left, and silk on the top right. You can also use 11 Essence in the top left, and fill the rest of the rod slots with will runes.
Tempers do almost nothing for mobs below level 100, but for level 90 players, you are starting to use your full potential of the tempers at level 105 mobs, and level 100 heroic mobs.
A good way to test if you're at the threshold for the zone for tempers is to use a weapon with a proc like Masq Staff. If the staff proc is not hitting for the full damage (500), then you need more tempers.
Sadoul recommends raising all 3 types. Once you're high enough level, you're going to want to use sloth aura. The magic tempers will help that to land more often.
Physical Essence are physical hit chance and miti penalty.
Magical Essence are for hit chance for both dot/ddd, and miti penalty.
Defense Essence are to add defense to you, but only up to a certain amount per zone. Once you hit the zone requirement for essence, they don't really help beyond that point for the zone.
Click the archery symbol in your inventory to view all three temper types.
OUT OF DATE - The formula for PVE tempers is % = 0.3x^(0.5) where x is the number of tempers you have.