Short answers, pulled from the story.
Dice notation is a system for describing combinations of dice in wargames and tabletop role-playing games using algebra-like expressions such as d8+2. It is also known as dice algebra, common dice notation, or RPG dice notation.
3d6 means rolling three six-sided dice and typically adding the results together, though some systems instead take the best single result. In the general form AdX, A is the number of dice and X is the number of faces on each die.
Ted Johnstone introduced standard dice notation in his 1975 article 'Dice as Random Number Generators,' published in the first issue of the fanzine Alarums & Excursions. Gary Gygax later built the notation into the first edition of Advanced Dungeons & Dragons, published between 1977 and 1979.
Percentile dice, written as d% or using X equal to 100, is most often rolled with two ten-sided dice, one representing tens and one representing units. A roll of zero on both dice is read as either 0 or 100 depending on the game.
A D66 roll means d6x10+d6, using one six-sided die for the tens digit and another for the ones digit to produce a two-digit result. It has 36 possible outcomes ranging from 11 to 66, with an average result of 38.5 and a standard deviation of about 17.16.
Exploding dice, also called open-ended rolling or penetration rolls, let a die be rerolled when it lands on its highest face, with each new roll added to the total. Shorthand for this includes adding an exclamation point after the roll, an asterisk, or doubling the final letter, and the mechanic appears in games like Feng Shui and Savage Worlds.