Equal number or higher means you capture the other card.

Elements get a +2 bonus on non elements.

Dark gets a +1 bonus on all elements but light.
When Dark defends against an element, it gets a +2.

Ice->Fire->Earth->Ice get a +3 bonus on eachother

Light gets a +3 on dark

At the end of the game, whoever wins, gets to pick an opponent card currently under their control/color

Credit to Bug

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Attacker v Defender Bonus

Neutral v Neutral 0 / 0
Neutral v Fire 0 / 2
Neutral v Grass 0 / 2
Neutral v Ice 0 / 2
Neutral v Dark 0 / 2
Neutral v Light 0 / 2

Fire v Neutral 2 / 0
Fire v Fire 0 / 0
Fire v Grass 3 / 0
Fire v Ice 0 / 3
Fire v Dark 0 / 2
Fire v Light 0 / 0

Grass v Neutral 2 / 0
Grass v Fire 0 / 3
Grass v Grass 0 / 0
Grass v Ice 3 / 0
Grass v Dark 0 / 2
Grass v Light 0 / 0

Ice v Neutral 2 / 0
Ice v Fire 3 / 0
Ice v Grass 0 / 3
Ice v Ice 0 / 0
Ice v Dark 0 / 2
Ice v Light 0 / 0

Dark v Neutral 2 / 0
Dark v Fire 1 / 0
Dark v Grass 1 / 0
Dark v Ice 1 / 0
Dark v Dark 0 / 0
Dark v Light 0 / 3

Light v Neutral 2 / 0
Light v Fire 0 / 0
Light v Grass 0 / 0
Light v Ice 0 / 0
Light v Dark* 0 / 0
Light v Light 0 / 0

* Note that Light v Dark may be bugged. In the current iteration of the game, the bonus is 0/0. It is not confirmed if this is intended or not.

Credit to ExplosiveDruid

If you want to see some videos on it:

9th Dawn III - Getting started with Fyued
https://www.youtube.com/watch?v=s0fZRGuFbCo

9th Dawn III - Playing Fyued from behind
https://www.youtube.com/watch?v=h-RHfEHI79M