Which countries can participate in the Olympic games? Not to participate in a particular sport (like this question asks) but to participate in the Olympics at all. Is there any criteria for a country to participate in the Olympics?
In order for a country to send a team to the Olympics, the country needs a National Olympic Committee (NOC) that is recognized by the International Olympic Committee (IOC). There are 204 recognized NOCs, and Wikipedia has a list of all the recognized NOCs, sorted by the date the NOCs formed.
Under the current IOC rules, which were changed in 1996, new NOCs will only be recognized from nations that are recognized as an independent country by the United Nations. However, before 1996, the IOC allowed NOCs from countries that were dependent territories of other nations, and these NOCs are allowed to continue. Therefore, in the Olympics, you will see athletes from countries like Bermuda, Puerto Rico, and Hong Kong, which are all territories of other nations. Currently, there are only three UN recognized countries that do not yet have NOCs: South Sudan, Vatican City, and Niue.
There are unofficial NOCs that are not recognized by the IOC from various territories and groups of people, and Wikipedia lists some of those. Some of these participate in the Paralympic games, but under the current rules they are not eligible for inclusion in the Olympics.