The term amateur is not unique to boxing. Most professional sports you can think of also have amateurs. What distinguishes them from the professionals is that they do not get paid for what they do. The highest honour in the amateurs is an Olympic Gold.
In boxing, depending on the country, a boxer has a set amount of time they can stay in the amateurs. At some point they reach a certain age or skill level and they have one of two options - quit or turn pro. You can look at a good proportion of professional boxers and dig up their amateur careers.
Take Mike Tyson for example, he had an outstanding amateur career (and a professional career as well). The clip above is from a 1982 Junior Olympic Championship. Mike is defending his title from the previous year.
Rules in the amateurs are different than in the pros. The scoring system is different - often under dispute, and safety is enforced above anything else. As a result, there are very few injuries in the amateurs compared to the pros. It is more common for a fight to be stopped by the referree than it is to see a knockout. (unless of course you're fighting Tyson as in the clip above).
There are some who propose to make the amateurs more like the pros, removing things like requirements for boxing headgear and changing the points system. The argument against headgear is that it doesn't protect against knockouts, only cuts and scrapes. While true, the amateur powers that be have made it pretty clear that isn't going to happen anytime soon.
There is a perception that headgear prevents brain injuries. While not necessarily true, the perception is beneficial in helping parents get over the initial shock of young Jimmy or Suzie coming home and saying they want to start boxing. The perception is that headgear somehow makes it safer. In reality it's the rules and attention to safety amateur refs enforce.
What are your thoughts - should the amateurs more closely resemble the pros in both scoring and conduct?