The USA is comprised of incorporated and unincorporated territories. The incorporated territories are Washington D.C. plus the 50 states. The unincorporated territories are Puerto Rico, the USVI, American Samoa, Guam, and the Northern Marianas. Puerto Rico happens to be the largest and most populous of those territories with a population that is larger than the population of 22 states so it gets the most attention.
Why these territories weren't incorporated into the union is a complicated topic that involves colonial politics and empire building so I'm not going to talk about it in this sub.
The main difference between the states, D.C., and the unincorporated territories is the level of government representation. The gist of it is that the unincorporated US territories fall under US jurisdiction but they don't get to participate in the US government. Washinton D.C. gets some representation because they at least get to vote for the US president.
Puerto Rico has a local government with its own constitution, both of which fall under the control of the USA. Obama is the president of Puerto Rico, the US dollar is the Puerto Rican currency, the armed forces of the USA are the armed forces of Puerto Rico, the US Postal Service is the postal service of Puerto Rico, the US federal agencies are the Puerto Rican federal agencies, and Puerto Ricans are natural born American citizens.
There is no difference between the US citizenship of a Puerto Rican and the US citizenship of someone from the incorporated territories. Whatever differences exist are the result of residence. For example, if a person from California becomes a resident of Puerto Rico he can no longer vote for the US government because Puerto Rico doesn't get US government representation. If a person from Puerto Rico becomes a resident of California he can vote for the US government because California gets US representation. American citizens don't need passports or visas to move between the US territories whether they are incorporated or not.
Puerto Ricans don't pay federal income taxes but we pay every other tax like import and export and payroll and state taxes. We could talk further about this issue but again, I think that it falls outside of the scope of this subreddit.
Puerto Rico has independent representation in the olympics and in Miss Universe and in other events because those events are private events by private companies and private companies get to determine who gets representation. The political parties of the USA are also private companies (sort of) so Puerto Ricans get to vote in the presidential primaries.
About 5% of the population of Puerto Rico wants independence, which is actually low when compared to the states of the USA, and the rest of the numbers are divided down the middle between those who want statehood and those who want to preserve either the status quo or some modified version of it. Referendums are held and people like to proclaim that somebody won but the reality is that no change is going to happen unless there is some form of consensus because no US politician wants to deal with the consequences of imposing a new status definition on 3.6 million US citizens.