The NCAA is the governing body for college athletics. Compared to Europe, it is the federation that determines the rules and regulations for each sport. However, the NCAA has clear and strict rules on both athletic eligibility and academic eligibility. For you to be eligible to play college sports you need to meet minimum academic requirements and "amateurism" status, which proves that you are and have been competing non-professionally.
Each university within the NCAA system is divided into a specific division. The NCAA sponsor Division I, Division II and Division III. The determinants of which school are within a given division is mostly based on school size, funding availability, athletic emphasis and school reputation. There is no promotion or relegation between the divisions. It is false to assume that every school in Division I is better than every school in Division II. Many schools between divisions overlap in strength of sport and total resource. The NCAA sponsors a total of 90 different sports for both men and woman. Each Division has it's own specific rules and regulations, hence it is important to make yourself familiar with the minimum requirements for your desired division. For example, in Division III no athletic scholarships are allowed to be given out, hence most Division III schools are more inclined to provide academic scholarships.
Within the NCAA, each school is associated with a conference. A conference is comparable to a league system in Europe. These conferences are predominately grouped by region to limit travel distance for competition. Each conferences mostly has between 6-20 members that compete against each other within conference matches. Every conference has one automatic qualifier for the NCAA National Tournament. You can view the conference games as league games that determine if you take part in the national playoffs.
At the beginning of the season, before conference starts, most sports schedule non-conference competition. This can be through individual matches or tournaments. The purpose of these non-conference competitions are to gain experience competing and most importantly build your national ranking within your sport. In the US there is a ranking for almost anything you can think of and people like to always have a relative comparison measure.
Non-conference competition is always scheduled by the individual institution and can include any university. Hence, non-conference scheduling is important because in some sports it can increase your chances in taking part in the NCAA National Tournament at the end of the season. If you compete and win against higher ranked opponents your ranking will increase. Institutions that are usually ranked high, logically schedule similarly high ranked opponents to receive the reciprocal benefit. Therefore, it is important you are in a university that allows for a competitive non-conference, if not conference schedule.
Every NCAA sponsored sport finishes the season with the NCAA National Tournament. You might have heard the terms "The Big Dance®", "March Madness®" and "College Football Playoffs®". Each teams goal and dream is to make it to the national tournament and to compete against the best universities in the nation. In most sports, there is one automatic qualifier through the respective conference competition and the remaining field is selected by the NCAA Selection Committee. This is called an "at large bid". The selection is mostly based on the previously mentioned rankings. Each winner of the national championship receives the legendary NCAA champion trophy and NCAA champion ring.