Unless stated otherwise, assume rules are as given in d20. I recommend The Hypertext d20 SRD, as it contains all the necessary foundation material. D&D 3.5 will also work.

DM Fiat

The first rule of roleplaying is that the DM can do no wrong. I'll occasionally rely on DM-fiat in order to speed the game, rather than bogging down in rules minutia. A few DM guidelines to help this along:

Tough or Challenging?: Most DCs are in the range of 15 (Tough) to 20 (Challenging). So, most of the time, picking one of these two DCs will have you decently covered. (See Skills Checks for more.)

"Roll first, ask questions later": Frequently the difference between DC15 or DC20 won't actually make a difference because of the roll. So roll first, and then decide what the DC should really be only if you have to. This is easier if you don't tend to name DCs to your players. Instead, ask what they rolled and then give the appropriate response.

1 minute lookup rule: If you can't remember a DC or effect, ask if anyone else can. If anyone really cares, see if they can do a lookup and recitation within a minute. If not, just call it and make a note to yourself (including what you thought the rule probably was) and look up the details later.

Core Mechanic

E6 Rules

Basically, level 6 is max level, after which PCs gain another feat after every 5000 XP. For full rules, see E6 Sourcebook.

Automatic Success and Failure

For non-skill rolls, a natural 20 roll means the attempt is at least barely successful, regardless of modifiers. (It is not possible to succeed on some near-impossible skill check just because you rolled a 20, though.) If the roll would have already been a success, it might mean some boon or benefit, such as a critical threat in combat.

A natural 1 roll means the attempt is at least a mild failure, regardless of modifiers. If the roll would have already been a failure, a mishap (such as a dropped or broken tool) may occur. This is largely dependent on the DM and situation, but some sort of save (such as a DC 10 DEX check) may still let you avoid the mishap.

Heroic Exertions

Once per encounter, a PC can exert herself to change the effects of a die roll. If used before the results of a roll are announced, the player can reroll the die and take the best result. If used after the results (including for DM-made hidden rolls), this can be used to change the results by one degree of success in either direction--for example, change a failure to a success, a critical failure (natural 1) to a regular failure, or a critical hit (natural 20) to just a regular hit. This can be applied to any roll the PC makes or to an oppenent's roll when in a contest with the PC (AC counts as contesting an attack). So a PC could make their attack successful, but also turn an oppenent's successful attack or save into a miss or failure.

Each exertion causes 1 damage to the most relevant stat (DEX for AC) and grants the GM one karma point.

The GM starts the game with one karma point. A GM can spend a karma point to grant any NPC a heroic exertion or to negate the effects of a PC's exertion, allowing the dice to determine the effect after all (the PC still suffers the stat damage for a negated exertion).

The GM may also allow other benefits or uses for exertions and karma points. For example, the DM might also use a karma point to introduce some extra complication or force a PC to fail a successful save. Or a magic-user might be able to use a mental exertion (possibly with some Wound Points) to regain an expended spell slot.

Checking: Skipping Rolls

Take 20 and Take 10 rules, as normal.

Default passive rolls: The DM can assume characters Take 10 (if they're paying attention) or Take 5 (if they're not) on their Spot and other passive checks. Thus, even the fact of making a roll is not revealed. The DM can still introduce some randomness here by instead rolling for the DC side, treating the DC-10 as a modifier against 10 + PC's skill modifier.

Group rolls: The DM can roll once for the whole group, and then apply each person's modifier to that roll see who succeeded and who failed. (This works nicely for speeding initiative checks, group Spot checks, etc.)