How does the term "American" changed from the founding of the nation through today?

America was named originally after Amerigo Vespucci who created the first maps of the new land. Since then, the name has transformed dramatically over the centuries. The initial term "American" was used to refer to the British colonists inhabiting the 13 colonies on the East Coast of the United States—the people who would eventually go on to lead the revolution and become the United States of America.
Over time, when the colonies became a true nation, the name took on the meaning encapsulating everyone in this new nation of America, and slowly other people joined in—from the Spaniards down in Florida and on the West Coast to the French who had come down through the Midwest after establishing fur trading in Canada.
Now, the term "American" identifies the people of the United States, and it encapsulates the nature of freedom and inclusion, as well as a melange of beliefs and viewpoints that have come into the nation.

The term American was used before the United States was even the United States. It was used to refer to the British Colonists living in North America. After the Declaration of Independence and the Revolutionary War, there was some mixed usage of the word American as the noun to refer to citizens of the United States. In the Constitution, the Founding Fathers referred to themselves as “We the People of the United States.” One of the first usages of the term in historical documents was in the Federalist Papers, authored by Alexander Hamilton and James Madison in the late 1780s.
Arguably the most important usage of the term in the beginning was by George Washington in his 1796 Farewell Address, in which he encouraged his fellow citizens to embrace the term and think of themselves as Americans. He said:

The name of American, which belongs to you in your national capacity, must always exalt the just pride of patriotism more than any appellation.

However, back during the founding days of the country, the term American did not necessarily mean anyone who lived in, or had been born in, the United States of America. Without a doubt, the term applied to white men—they were the “we” in the opening line of the Constitution and were the only citizens at that time to have the full range of rights that all Americans now have, legally speaking. Women probably were thought of as Americans, since they were citizens (although they could not vote). It is doubtful that slaves, even those born in the United States, would have been considered Americans as they were not considered to be citizens.
Throughout the years, the term has come to encompass connotations that are both positive and negative. During the late 18th and early 19th centuries, America came to be known as a country of freedom and a beacon of hope for immigrants around the world, especially those coming from areas of famine-stricken Europe. As the 20th century progressed, the term American had diverse, at times polarizing, undertones. WWII generally showcased the strength and resolve of Americans, whereas eras such as the Cold War, the Vietnam War, and wars in the Middle East have somewhat tarnished the view of Americans by people from other countries.
Works Cited
"Constitution of the United States - Official," National Archives and Records Administration, http://www.archives.gov/exhibits/charters/constitution.html.
"President George Washington's Farewell Address". Ourdocuments.Gov, 1796, https://www.ourdocuments.gov/doc.php?flash=false&doc=15.
"The Complete Federalist Papers". American History, From Revolution, To Reconstruction And Beyond, http://www.let.rug.nl/usa/documents/1786-1800/the-federalist-papers/.