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, 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 a 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.