A cheat-sheet to new players quickly understand elements at a glance.
Guide to Cheat the Elements
Some Notes on Various Elements
Burning enemies steadily take damage over time. It's straight-forward enough, but this makes fire particularly beneficial for groups of fodder enemies without shields or armour - a single proc of Burn can deal a great deal of damage to a target without you even needing to put yourself into line of sight of it.
Shock hits the target with a small area-of-effect zap and inflicts them with a debuff that causes them to take 10% more damage from all sources.
10% is more than it sounds, that's roughly equivalent to your weapon's damage being +1 level higher against Shocked targets.
Decay decreases the target's movement speed by 50% - the immediate observation is that this is particularly good for melee targets that rush you down - keeping them away from you and keeping the pressure off.
Another important part of Decay's value is any target at all that moves - even if it's not particularly fast - will be much easier to track when slowed, indirectly boosting your damage by making it easier to hit their head other crit spot. This is especially apparent on projectile-based corrosive weapons such as Aura of Venom and Prism.
Crown Prince (that's the cat boy) has smoke grenades that deal powerful Corrosive damage that makes for a very easy way to proc Decay in a small area. To make the most of these benefits, prioritize Ascensions such as Acidominance and Elemental Rage to greatly improve your effective damage from smokes, Backpack Expansion to boost how often you can spam them, and at least one rank of Hex Smoke to further debuff targets.
A target inflicted with both Burn and Shock suffers from Manipulation. Manipulation, at least on paper, turns the target friendly and makes them start attacking their allies rather than you. The AI doesn't exactly seem great for this - sometimes they don't attack at all and I don't really tend to notice much damage coming from them. It'd be easy to write this off as a pretty useless fusion, but the key words to pick up from its effect would be the words 'rather than you'.
The power of Manipulation comes not from them fighting their allies, but from them not fighting you. This gives a great reprieve from more threatening foes and buys you precious time to clear fodder, reposition, or just rinse through their health without needing to worry about dodging their attacks.
A target inflicted with both Burn and Decay gets hit with an Explosion. Unlike all of the other effects and fusions, Explosion is not a status but rather a single proc of an area-of-effect damage tick.
The power of Explosion, Burn being particularly effective on groups of unshielded and unarmoured targets and Decay having an AoE proc right in Crown Prince's kit with various associated buffs means that combining fire and corrosive damage is incredibly strong when dealing with groups of enemies. Acidic Explosion and Hex Smoke are Ascensions that add even more damage and AoE, allowing fodder to be cleared far far into the game even with scrolls and weapons that aren't ideal for it.
A target inflicted with Shock and Decay is inflicted with Miasma, and Miasma is the unparalleled king when it comes to straight damage per second.
Miasma deals a relatively low damage damage-over-time effect on its target, which spreads to nearby enemies and gives them an even lower damage over time effect. However, this damage stacks - you can repeatedly apply Miasma on a target and its friends over and over (I've read that it caps at 9 times). And once you've stacked Miasma a few times, it is an absolutely ridiculous amount of damage that utterly shreds through health.
Many players prioritize their builds to proc Miasma as much as possible - a common goal as Crown Prince is to find a reliable Lightning weapon to combine with his corrosive grenades and taking whatever scrolls, inscriptions and Ascension that maximizes the chance and effect of getting both Shock and Decay onto as many targets as possible.
Miasma is almost a better version of fire - on top of inflicting both Shock and Decay, it deals greater damage over time than Burn and frequently better area-of-effect output than Explosion. While the playstyle you should prioritize depends heavily on both your weapons and the various ascensions and scrolls made available to you, setting your sights on Miasma early on is rarely a bad idea.