Whenever you attempt an action that has some chance of failure, you roll a twenty-sided die (d20). To determine if your character succeeds at a task you do this:
- Roll a d20.
- Add any relevant modifiers.
- Compare the result to a target number.
If the result equals or exceeds the target number, your character succeeds. If the result is lower than the target number, you fail.
Dice rolls are described with expressions such as "3d4+3", which means "roll three four-sided dice and add three" (resulting in a number between 6 and 15). The first number tells you how many dice to roll (adding the results together). The number immediately after the "d" tells you the type of die to use. Any number after that indicates a quantity that is added or subtracted from the result. Sometimes you add several dice together that aren't all the same number of sides, such as "1d8+1d10", but it's the same sort of thing.
d%: Percentile dice work a little differently. You generate a number between 1 and 100 by rolling two different ten-sided dice. One (designated before you roll) is the tens digit. The other is the ones digit. Two 0s represent 100.
In general, if you wind up with a fraction, round down, even if the fraction is one-half or larger. The only time you should round up is when the formula in question explicitly says so, in which case you do so even when the fractional part is less than 1 (such as "one third your level (rounded up)" being a result of 1 at 1st level). Certain rolls also have a minimum of 1, even when you round down.
Sometimes a rule makes you multiply a number or a die roll. As long as you're applying a single multiplier, multiply the number normally. However, when two or more multipliers apply to any abstract value (such as a modifier or a die roll), combine them into a single multiple, with each extra multiple adding 1 less than its value to the first multiple. Thus, a double (x2) and a double (x2) applied to the same number results in a triple (x3, because 2+1=3). If it helps to make sense of it, keep in mind that each doubling is intended to represent +100%, and so when combined two "doubles" (2 * +100%) is a "triple" (+200%). Similarly, higher multipliers or additional multipliers use the same concept. A x4 and a x3 are a x6 when combined (4+2), and three x2s are a x4 (2+1+1).
When applying multipliers to real-world values (such as weight or distance), normal rules of math apply instead. A creature whose size doubles (thus multiplying its weight by 8) and then is turned to stone (which would multiply its weight by a factor of roughly 3) now weighs about 24 times normal, not 10 times normal. Similarly, a blinded creature attempting to negotiate difficult terrain would count each square as 4 squares (doubling the cost twice, for a total multiplier of x4), rather than as 3 squares (adding 100% twice).