From the MAA website:
About AMC 10/12
The AMC 10 is a 25-question, 75-minute, multiple choice examination in secondary school mathematics containing problems which can be understood and solved with algebra and geometry concepts. The AMC 12 is a 25-question, 75-minute, multiple choice examination in secondary school mathematics containing problems which can be understood and solved with pre-calculus concepts. Two different versions of the contests are given on two dates, about two weeks apart, in February. Each AMC 10 and AMC 12 contest contain about 12 of the same math problems.
Purpose of the AMC 10/12
The main purpose of the AMC 10/12 is to spur interest in mathematics and to develop talent through the excitement of solving challenging problems in a timed multiple-choice format. The problems range from the very easy to the extremely difficult. Students who participate in the AMC 10/12 should find that most of the problems are challenging but within their grasp. The contest is intended for everyone from the average student at a typical school who enjoys mathematics to the very best student at the most special school.
A special purpose of the AMC 10/12 is to help identify those few students with truly exceptional mathematics talent. Students who are among the very best deserve some indication of how they stand relative to other students in the country and around the world. The AMC 10/12 provides one such indication, and it is the first in a series of examinations. The AMC 12 is one in a series of examinations (followed in the United States by the American Invitational Examination and the USA Mathematical Olympiad) that culminate in participation in the International Mathematical Olympiad, the most prestigious and difficult secondary mathematics examination in the world. In this way the very best young mathematicians are recognized, encouraged and developed. Another valuable comparison students can make is between their own level of achievement and their levels in previous years. In particular, they are encouraged to begin taking the contests early in their mathematics studies and to look back with pride each year on how they have learned to answer questions that they could not have answered previously.
For more information, see the MAA's website for the AMC 10/12.