This is the first of three articles discussing common dice-using mechanics used in tabletop RPGs (aka "narrative" or "pencil-and-paper" RPGs). This articles covers how dice mechanics are generally used, and in light of this what criteria they should be judged on.
Nearly all dice mechanics share a similar concept at the heart. To resolve an attempted action, the character has a numerical rating of her effectiveness: here called the "stat". There may sometimes be two stats (generally "attribute" and "skill"), but more often these are combined into a single number. The action also has a numerical rating of "difficulty", which may default to a standard. Dice are then rolled to determine whether the action is a "success" or a "failure", and the degree of success/failure.
Dice abbreviations are "d" followed by the number of sides. i.e. "d6" is a six-sided die and "d10" is a ten-sided die. A number before the "d" indicating to roll that many dice. Thus, "4d6" means rolling four six-sided dice. Unless noted otherwise, assume the numbers are all added together. The dice commonly available at hobby stores are: d4, d6, d8, d10, d12, and d20. "Percentile dice" (aka "d100" or "d%") refer to a special roll from 1 to 100, where you roll one ten-sided die for the "tens" digit and another ten-sided die for the "ones" digit.
"Open-ending" refers to mechanics where if a certain roll of the dice comes up, you make another roll: re-roll and add, roll extra dice, or something similar. This can allow for a theoretically infinite range of results. Open-ending can be added to any scheme of dice mechanic.
"Variance" is a general term for how widely spread out the results are. For example, rolling 1d6 and taking the highest roll out of 5d6 both can produce results between 1 and 6. However, the highest out of 5d6 has much less variance. This can be measured by the "root-mean-square" of the distribution, which is calculated by:
|RMS= SquareRoot[ Sum over all rolls of (Result - Average Result)2 ]|
There are a number of semi-objective factors which dice mechanics can be judged on. I will outline a few of these below.