I looked around the site, too, and there doesn't seem to be a good, high-level overview.
Getting the Distinguished badge through the CMP is achieved by earning 30 total points in EIC (Excellence in Competition) matches. Points are awarded to the top 10% non-distinguished shooters in a match, and the number of non-distinguished shooters competing determines how many awards (called "legs") are given out. For instance, a match with 40 non-distinguished shooters would have four legs awarded - one gold (10 points), one silver (8 points), and two bronze (6 points). Also, you must have one 8- or 10-point leg to earn the badge. You can compete in up to four matches per year and be eligible for points - any three sanctioned matches and the national championships at Camp Perry.
The rule book has all the details, including specs on what types of rifles and pistols can be used.
http://www.odcmp.com/Competitions/Rulebook.pdf