A semi-professional athlete is one for whom sport is not a full-time occupation. They are not amateur because they receive regular payment from their team (company), but at a much lower rate than a full-time professional athlete. As a result, players may have (or seek) a second full-time job. A semipro player/team could also be one that represents a place of employment that only the employees are allowed to play on. In this case, it's considered semipro because their employer pays them, but for their regular job, not for playing on the company's team.
When applied to vocational tools and equipment, it refers to products that lie between the amateur and professional levels in both quality and cost, though nowadays the term prosumer is often used instead.
The San Francisco Olympic Club fielded an American football team in 1890.  That year, the Olympic Club was accused by a rival club of enticing athletes to jump to its ranks with offers of jobs. An investigation by the Amateur Athletic Union ruled that the Olympics' practice was not actually professionalism but only a "semi" form of it, inventing the term "semi-pro". Although the Amateur Athletic Union did not like the idea very much, it decided that clubs could indeed offer employment without losing their amateur status or compromising the athlete. 
In North America, semi-professional athletes and teams were far more common in the early and mid-20th century than they are today. There are many benefits, such as collegiate eligibility and the attendant scholarships, in maintaining amateur status (unlike the Amateur Athletic Union, the NCAA forbids any sort of compensation outside of scholarships, including job offers tied to their playing). Eligibility for participation in the Olympics in some sports is still dependent upon maintaining a purely amateur status (although far less so than was previously the case), and such athletes may be supported by government money, business sponsorships, and other systems. At the same time, professional sports have become such a massive and remunerative business that even many low-level feeder teams can afford to have fully professional athletes.
Semi-professionalism is most prevalent in junior ice hockey, in which the top levels of Canadian (and European as well) junior hockey (most of whom are teenagers still in, or just out of, high school) are paid at a semi-professional level. This is not the case in the United States, where college ice hockey dominates at that age group; the junior leagues in the United States generally operate as fully amateur teams to maintain the players' eligibility to play in college.
Lower-end minor leagues and more obscure sports often operate at a semi-professional level due to cost concerns. Because the cost of running a fully professional American football team is prohibitive, semi-pro football is common at the adult levels, particularly in the indoor variety, providing an outlet for players who have used up their NCAA eligibility and have no further use for maintaining amateur status; as a sport that normally plays only one game per week, football is especially suited for semi-pro play. The National Lacrosse League, whose teams also typically play only one game per week, pays a salary that is enough to be considered fully professional, but players also are able to pursue outside employment to supplement their income.
There are several hundred semi-professional football teams at non-League level. The bottom division of The Football League (the fourth tier of the English football league system) has traditionally been the cut-off point between professional ("full-time") and semi-professional ("part-time") in English football. However, many teams in the top non-League competition, the National League have become "full-time" professional clubs in an effort to achieve League status.
Women's football in England is semi-professional at the top levels, as finances depend on promotion and relegation both of parent male teams and of the female teams themselves. Full professionalism for women is still in the planning stages; top female players often depend on other sources of income (such as coaching and physical training), and many attend university or college while playing.
Historically, English rugby league and rugby union have had one full-time professional division, with semi-professional divisions at the next level down. The second tier of union, the RFU Championship, became fully professional beginning with the 2009–10 season.