After significant delay, I figured it was time to post the actual rules--how you accomplish things--in Anti-d20. In general, there are three basic types of rolls that you need to make in Anti-d20, but there are also some optional rolls that can come up.
1) The first type of roll is called a Check, as in "Give me a Strength check." Checks can be made against traits or skills. To make a Strength check, look at the Strength trait, see what type of dice it is, and roll that dice. You need to meet or beat the Target Number with this check. The default Target Number is 4, but of course, the GM can assign any Target Number that he thinks is appropriate given the difficulty of the task. Using a Strength Check to catch and hold a falling steel portcullis might be quite a bit more difficult than 4.
The GM is also responsible for adjudicating what happens on a success or a failure. On the example above, the GM might assign a difficulty of 8 or even 10 to catch a portcullis. A success means that the character has managed to halt the fall of the portcullis and the group can sneak through before he lets it fall again. A failure might mean that the portcullis crashes into the character, crushing him and calling for a Health Roll.
2) An Opposed Roll is one where two characters (player characters or non-player characters) make opposed rolls; i.e., instead of a Target Number, to be successful, one character must beat the check rolled by another character. A Perception Trait check vs. a Sneak Skill check would be one example. Two characters in melee combat would make Opposed Rolls using their Fighting Skill. (A character who wants to shoot another character probably also rolls against the other characters Agility Trait assuming they attempt to dodge out of the way, modified as necessary by the GM. For example, for shooting a modern firearm at close range, I'd give the defending character's agility roll a significant penalty; maybe -6 or even more.) Again, the GM is responsible for defining what success and failure mean. In melee combat, it could mean that the losing character has to make a Health Roll, for instance. In a Perception vs. Sneak check, obviously the Perceiver notices the other character if he wins the Opposed Roll and does not if he loses it.
3) The third and final of the basic rolls in the game is the Health Roll. This is what happens when something bad happens physically to a character, and the Health Check is made using the Health Trait. Both the Target Number and the results of a check, either successful or failure, are up to the GM and are highly situation dependant. Getting punched in the face and failing a Health roll might only stun the character for a moment, while getting shot in the face with a shotgun will have a much higher Target Number, and failure might mean instant death.
As a general rule, as with other Checks, a Health Check baselines to a target number of 4, but could fluctuate considerably based on circumstances. As a very general rule, failed Health Checks could temporarily reduce the Health trait by a step on the dice scale. So a character with a d8 Health skill could temporarily go down to d6. This, however, represents relatively minor damage, and any given situation could be considerably worse.
In general, the more steps you lose, either cumulative or through a single devastating failed Health roll, the more difficult it is to recover, and the more recuperation is needed, as well as possibly first aid or protracted medical care.
This isn't really handled in the rules, though. There's a Heal skill, and GMs can adjudicate based on what they think is appropriate. Because what is appropriate is as much based on the genre as it is on anything realistic, then that can vary considerably.
Lastly, there are some optional rolls. The Luck trait can call for a Luck Check, which the GM can call for as a "safety net" for catastrophically failed rolls. Characters never announce their intent to make a Luck Check; it can only be used when the GM says it is appropriate. It doesn't change a failure to a success, but it could change a catastrophic failure into a more routine one.
And Sanity, for games that use it, work very much like Health checks, except, naturally, for mental rather than physical health.