I would say that it is extremely important to remember that this question is purely subjective and each response to this question or other types of this question will vary depending on who you ask, even within America. Going back to the question, "What does it mean to be an American?", I believe it comes down to a few key ideas that make up the core of American identity.
Assimilation
Being an American means assimilating into American life while still knowing your roots. Certain people may see this one as a controversy, but that mostly comes with the way they think about it. I feel that I have good grounds to talk about this particular core point as my family came to America as immigrants from another country out of eastern Europe. I hear plenty of people say that America is a "Melting Pot" and that America is a country of diversity. While true, this statement is often misunderstood or mis-explained.
America is a "Melting Pot" in the terms of cooking a soup that has properly cooked the items in said soup that "Melts" all the items together in a way that tastes amazing, hence, the word "Assimilation." America is not a breeding ground or "food" experiment for the entire world to flood into that makes the soup have uncooked items that do not fit well with the soup that is already there. You do not add Cayenne pepper seasoning to a soup that does not need spice. People may not like this, or even understand this concept.
I have no issue with people coming to America who become a citizen and follow the laws that are in this country and properly assimilate—my parents did this, and so have many others in this country. That said, there is nothing wrong with having your culture in America, or you as a person, but the way you treat America as you reside in it. You do not go to someone else's house and demand that they give you anything, force the way that you lived elsewhere on to that person, or demand that you change the laws of the land that are already in place to the laws that you had where you came from. This is that spice. Without spending too much time on this one point, Assimilation is a core factor in what it means to be an American because it reflects how you treat America when it gives you a chance.
Being Proud to Be an American
In addition to assimilation, pride also plays a central role in what it means to be an American. Once again, this one may be controversial, but I feel that being proud to be an American is quite important to actually being an American. Being proud to be an American does not mean that you are proud of your government as some people may think. Thinking that way is not very forward thinking. You can still be proud to be an American, but disagree with the government.