As the United States is so large and comprised of so many various immigrants, people often refer to their heritage. Americans pride themselves on the "meting pot" culture, as it has always been a big part of the strength and adaptability of their country as a whole. So an American might say that they are Italian, Polish, Russian or Irish - they do not mean that they are those nationalities, this is usage within the context of America.
Hyphenated Americans tend to have cultures slightly different from mainstream American culture; habits and traditions brought over from the countries their family emmigrated from. Many retain the drinks, food, language, customs,and music of their heritage. They also tend to look like their cultural background...Italian, Irish, Indian, Polish etc... Some Europeans feel that they are aping the culture of European countries, but often they are just acting as they were brought up to act by their European born parents. Whether Irish-American, Russian-American, Italian-American etc...they all share some cultural characteristics with the countries their families left.It really is not such a big deal, all countries with heavy immigration do the same, such as Canada, England, Australia and New Zealand. Americans get the most flack for it though.
Tony Soprano (although fictional)is considered Italian/Italian-American when in America. He is an American first, but he is also from Jersey and is Italian American. Italian-American, his Hyphenated Americans group, is just a statement of his cultural background.