In North America, amateur baseball includes Little League Baseball, high school and college baseball, summer collegiate baseball, and various leagues in which adults compete and which are part of the National Baseball Congress. Semi-pro baseball is often considered to be part of amateur baseball nowadays, as even though some players may be paid small salaries or other forms of compensation to compete, this is not a means to earn a living. Basically, anything outside of organized baseball and the independent leagues is considered amateur.
In a lot of other countries, all baseball is considered amateur, even top-level national leagues, as the amount of compensation is minimal. Famously, Cuba claimed for a long time that the Cuban National League was fully amateur in spite of evidence that many players were earning a full-time living from playing, allowing its athletes to compete internationally when other countries had to field "true" amateur teams. In these countries, it is professional baseball that is the exception.
The distinction between amateur and professional sports used to be very important, particularly following the creation of the modern Olympic Games in 1896, as professional sports were considered tainted by their monetary aspect and their athletes were banned from the Olympics and many other national competitions, including collegiate sports as well as international competitions (like the Pan American Games, Asian Games, Baseball World Cup, Central American and Caribbean Games, etc.). This resulted in several controversies - the Mexican national team of the 1930s and 1940s routinely used Mexican League pros, resulting in their having to withdraw from the 1935 Central American and Caribbean Games. The Canadian national team had to forfeit a win in the 1967 Pan American Games when it was discovered that they had four players with pro experience. There are still certain limits enforced, e.g. a collegiate athlete cannot earn outside money for his sport and retain his eligibility, but these have been largely reduced as someone who plays minor league baseball for a salary can still compete in college football or basketball, for example. The Olympics threw in the towel in the 1990s, basically recognizing that top-level athletes must have a means of earning a living, and professionalism is no longer by itself a reason for exclusion. Court decisions in the late 2010s have further eroded the bans, with collegiate athletes gaining the right to monetize their image and likeness while retaining their amateur status.