Following the tables provided by GNAS (not available online for copyright reasons), each scored round is assigned a handicap level. For an archer to achieve his first handicap, he must first submit three scored rounds, after which he is given a starting handicap of the average of the three handicap levels for those rounds. So, say he scores handicap levels of 80, 70, and 60, his base handicap will be 70. From then on, his handicap will be improved every time he shoots a round with a handicap level better than his current handicap, to the average of his handicap and the handicap level of the "better" round. So our new archer:-
Shoots a round with a handicap level of 72 - no change
Then shoots a round with a handicap level of 68 - his handicap improves to 69 ((70+68)/2)
And so on. Your handicap never gets worse through the season, only better
At the end of the season, all handicaps are reassessed to the average of the handicap level of the best three scores submitted (There are a couple of added twiddles to use when insufficient scores have been submitted)
In a "Handicap Shoot", there are tables to assign an "allowance" for the archer's handicap level according to teh round being shot. So, say an archer with a handicap of 46 shoots a Warwick and scores 223. To assess his place in the handicap competition, the scorer adds the appropriate allowance (in this case 1114) and so his adjusted score is 1337. An archer shooting in the same competition could have shot a score of 53 in a New Western with a handicap of 70, and, with his allowance, would have an adjusted score of 1456, thus beating the original archer. They can be compared despite shooting different rounds and, in this real case from yesterday, different bowstyles.
Outside of competitions like this, handicaps are best used as a benchmark of an archer's progress. If you score a handicap level of 56 in a York, you are (theoretically) moving in the right direction from shooting a level of 66 in a Short junior Warwick. A very rough comparison, but one that can be made which, without handicaps, would be impossible.
Hope this helps. I am on standby for a flurry of further questions.