Circumstances or character aspects may influence your chances of success or failure on a check. These are represented with bonus and penalty dice, which are six-sided. They may be denoted as +#d or -#d, respectively.
When you make a check with bonus dice, if any of the bonus dice come up 6, you succeed, regardless of your d20 roll. Conversely, if you make a check with penalty dice and any of them come up 1, you fail, regardless of your d20 roll.
If you have both bonus and penalty dice, they cancel each other before rolling. For example, if you have +2d and -3d, roll just 1 of penalty die. You never roll bonus and penalty dice at the same time.
If any bonus die comes up 6 and your d20 check also succeeds, that is a critical success.
Likewise, if any penalty die comes up 1 and your d20 check also fails, that is a critical failure.
ConversionsIf something in your system would call for a +2 or -2 on a d20 roll, you can substitute one of these dice. They turn out to be about the same - the left column here is a target number, and the header is a character's modifier.
Those hit numbers work because the bonus die only matters if the d20 roll misses. The highlighted parts are where most rolls seem to happen, at least in the early levels of the game, or in a lower-powered one.
These dice become less important for skilled characters to succeed, but do help in avoiding mistakes and in driving crits. They reward less-skilled characters for seeking them out.
Almost everything in my game got converted into bonus and penalty dice, so the d20 rolls changed infrequently, since they were based on ability scores. You don't have conditional modifiers to tack onto the roll - they are all individual dice, with no math involved.
That also means there aren't something that would translate as conditional +1 modifiers, but I have never much liked them anyway.
Some character aspects could give bonus dice, but mostly they're situational. Charging? Bonus die for everyone! Attacking something invisible? Well, have a couple penalty dice.
Also note you can't crit or fumble without an extra die. You really want to seek out bonus dice for this reason, either to line op the crit, or negate the possibility of a fumble.
Adding more dice in either direction scales the chance of success or failure interestingly, though I think getting more than 3 in either direction would be rare.
I also ended up getting rid of a lot of level bloat by relying on these dice. If you were facing off against a thing of a substantially higher level, you got a penalty die, and it got a bonus die. Same thing if weaker opponents were facing you.