Attack Dice, and Attack Rolls, are made with a D20. If the result of that die is greater than the target's Defense, it is a hit. One Damage Die can be scored for each Attack that is a hit. So if a player had 2d20 Attack Dice, and 3D8 Damage Dice and both Attack dice are hits, then he may score 2 of the resulting D8 dice.

If the player has Accuracy, then the amount of Accuracy they have is added to each D20. So if they have Accuracy +2, and their 2 D20's roll a 10 and an 8, then the total results would be 12 and 10.
If a player has Advantage on their Attack, they roll an additional D20 and discard the lowest result.
If a player rolls a 20 on their Attack Dice then they get a Critical. For each bonus to their Critical, the required roll is reduced by 1. So if they have +2 Critical then they only need an 18 on an Attack Roll to get a Critical.
Attack dice are defined as a D20. Anytime someone makes an Attack Roll they roll a D20. Anytime an ability lists a number of D20 dice those are exclusively Attack dice. A d20 cannot be used for anything else.
If the result of a D20 is greater or equal to the target's Defense it is considered a hit.
When a player says they will roll a number of D20 dice the result is randomly generated and given to them.
If someone gets a Critical on their Attack they gain +1 to a Damage Die for each Critical they had during their Attack. So if they rolled 4D20 attack dice, got 2 Criticals, and roll 4d8 damage dice for results of 4, 5,6, and 7 they could apply +1 Damage to the 6 and 7 for results of 4,5,7, and 8.
If a player has an ability that has the category "Critical" they can make use of one of those abilities for each Critical they get during their attack.