I am a sprinter and there are 3 sprint events in the Olympics: the Team Sprint the Keirin, and the Match Sprints. The Team Sprint is a two man event and I currently do not have a partner so I have not been doing this event. The keirin is an 8 lap race with the first 5 ½ laps following a motor bike, and the last 2 ½ laps is a drag race to the finish line. Here is an example.
The Sprints consist of a 200m time trial which seeds the riders, and a series of match sprints where riders race against each other one at a time in a tournament. A match sprint example is here.
Based on the points acquired at the World Cups, World Championships and Continental Championships during the 2 years prior to the Games (12 events total), there are 10 countries that will qualify for the Team Sprint. One team per country. If your country qualifies in the Team Sprint you automatically qualify a spot in both the Sprint and Keirin.
For the Sprints and Keirin, in addition to the 10 spots allocated from the Team Sprint, there are 8 additional spots. If your country has qualified a spot with the Team Sprint, they are ineligible for the next 8 spots; only 1 rider per country in each event.
The last detail is that there is a maximum number of riders that can qualify per continent. For countries in the Americas, 2 countries can qualify in the Team Sprint and 2 more countries can qualify in the Sprint and Keirin.