There are two types of conflicts: a crewmember against Planet X and two crewmembers against each other. Every time a conflict is fought you have to compare two numbers each representing the success of a participant.
First, you should calculate how many dice you will use.
Duties: Decide whether your duty is relevant in this specific conflict or not. If it is, you have to decide whether your character uses rather practical or rather theoretical tools. If the former, take your ability value of the relevant duty. If the latter, take your knowledge value of the relevant duty.
If your character has more than one duties and more than one duties apply, add their values together.
It is imaginable that one duty is relevant in a practical way, while another in a theoretical way, so you can add ability and knowledge values, but not of the same duty.
That is the number of dice provided by your duty/duties.
Background: Decide whether your background is relevant. If it is, your background provides as many dice as its value is.
Shady Side: Decide whether your shady side is relevant. If it helps to succeed, it provides as many dice as its value is.
You got a pool consisting of dice provided by your duties, your background and your shady side.
Now you have to calculate the modifier applied to your roll.
Equipment: Decide what equipment is used by your character in this conflict. Add the value of any equipment. The sum will be the modifier.
The conflict is resolved by rolling dice. The result of your rolled dice and the modifier is added. The participant with the higher sum succeeds in the conflict, the other participant fails.
Using Earth
If the principle of Earth is relevant in the conflict and helps to succeed, you should add as many dice to your pool as the value of the principle is.
Using the Opponent's Shady Side
If you know your opponent's shady side, you can use it. If you are using it, you can add to your dice pool as many dice as the value of your opponent's shady side is.
Uncovering Someone's Shady Side
If you know a crewmember's background and you want to know his/her shady side, you have to start a conflict against him/her. In this case your opponent's background is always relevant to hide the shady side.
If you succeed, even then you won't know the specific truth, only get a hint. The clarity of the hint depends on the difference between the two die rolls.