The 107% Rule is a means to ensure that entries to a race are fast enough to be competitive, and not pose a danger to faster cars. It was originally enacted for the 1996 season, and was in place through 2002. The rule reintroduced for 2011, and has been in place ever since.
The rule as it stands now is fairly simple: during the first qualifying session, every car has to turn a time that is within 107% of the fastest time. For example, if the fastest car in Q1 turned a time of one hundred seconds (1:40.000), then all of the other cars would have to be faster than one hundred and seven seconds (1:47.000). Cars that fail to reach that threshold can appeal to the stewards, if they can show evidence that the car was clearly fast enough in timed practice, and the time in Q1 was caused by issues like weather, mechanical problems or an incident. But if a car was consistently slow all weekend, and failed to post a fast enough time, that car will not be allowed to start the race.
As shown by the chart below, it has been a number of years since the rule has had to be enforced.