Accelerated Reader, usually referred to simply as 'AR', is a common software program in American public schools. It is intended as a daily progress monitoring and assessment system for reading skills. It is widely used by elementary and middle schools, and is integrated with the Common Core Standards. It was developed by Renaissance Learning, Inc., which also has a number of related projects, for example Accelerated Math and the STAR assessments.
As far as the students are concerned, Accelerated Reader means that after they read a book they will take a short quiz on the computer that asks them some simple multiple-choice questions about the story. If the student is reading books of the appropriate level, these questions should be pretty easy, and it should be common for kids to get 100%. It is important to note that the 'appropriate level' is not their grade level, but the level at which they are successful.
For the teachers, this is a great way to get statistics on student reading rates, reading comprehension, and reading level. Accelerated Reader has tests for over 140,000 books, both fiction and non-fiction, and including a few selected textbooks and magazines (such as Weekly Reader). If a quiz is not already available, teachers can create their own.
In my experience, most kids find the program very motivating, and enjoy taking the quizzes; the kids are actively encouraged (and even required) to read books of the appropriate level, and therefore regularly get 80-100% on the quiz. Given that the students are also expected to read daily and that many students do not really enjoy reading for the sake of reading, AR is probably a good thing. However, it does put in place a demanding treadmill of constant reading, and has a default expectation of a student's reading level. If a good support system is not in place this may allow students to fall between the cracks. I am lucky enough to work at a wonderful school that gives lots of support to the students, and I have no complaints about the AR program.