The following quotations are exemplary descriptions of the notion that America is a melting pot of different ethnicities.  Don't take them as describing something that is substantially true, but rather as revealing certain kinds of attitudes that Americans have had about ethnic differences.  What attitudes are revealed here?  All of these quotations are taken from Werner Sollors' Beyond Ethnicity (Oxford UP), an excellent if difficult book that would be worth your time reading if you are interested in some of the complexities surrounding the concept of ethnicity.
 

Ubi panis ib patria, is the motto of all emigrants.  What then is the American, this new man?  He is either an European, or  the descendant of an European, hence that strange mixture of blood, which you will find in no other country.  I could point out to you a family whose grandfather was an Englishman, whose wife was Dutch, whose son married a French woman, and whose present four sons have now four wives of different nations.  He is an American, who, leaving behind him all his ancient prejudices and manners, receives new ones from the new mode of life he has embraced, the new government he obeys, and the new rank he holds,  He becomes an American by being received in the broad lap of our great Alma Mater.  here individuals of all nations are melted into a new race of men, whose labours and posterity will one day cause great changes in the world.
      --St. Jean de Crevecoeur

[The] triumph and general adoption of the english language have been the principal means of melting us down into one people, and of extinguishing those stubborn prejudices and violent animosities which formed a wall of partition between the inhabitants of the same land.
      --DeWitt Clinton, Governor of NY

The "Melting Pot" exercises were dramatic in the extreme:  A deckhand came down the gang plank of the ocean liner, represented in canvas facsimile.
 "What cargo?" was the hail he received.  "About 230 hunkies," he called back.  "Send 'em along and we'll see what the melting pot will do for them," said the other and from the ship came a line of immigrants, in the poor garments of their native lands.  Into the gaping pot they went.  Then six instructors of the Ford school, with long ladles, started stirring.  "Stir!  Stir!"  urged the superintendent of the school.  The six bent to greater efforts.  From the pot fluttered a flag, held high, then the first of the finished product of the pot appeared, waving his hat.  The crowd cheered as he mounted the edge and came down the steps on the side.  Many others followed him, gathering in two groups on each side of the cauldron.  In contrast to the shabby rags they wore when they were unloaded from the ship, all wore neat suits.  They were American in looks.  And ask anyone of them what nationality he is, and the reply will come quickly, "American!"  "Polish-American?"  you might ask.  "No, American," would be the answer.  For they are taught in the Ford school that the hyphen is a minus sign.

      --Description of the Ford Company English School Melting Pot rituals found in the Ford Times
 

I hate the narrowness of the Native American party. . . .  Man is the most composite of all creatures, the wheel-insect, volvox globator, is at the beginning.  Well, as in the old burning of the Temple at Corinth, by the melting & intermixture of silver & gold & other metals, a new compound more precious than any, called the Corinthian Brass, was formed so in this continent,--asylum of all nations, the energy of Irish, Germans, Swedes, Poles & (the) Cossacks, & all the European tribes,--of the Africans, & of the Polynesians, will construct a new race, a new religion, a new State, a ne literature, which will be as vigorous as the new Europe which came out of the smelting pt of the Dark Ages, or that which earlier emerged from the Pelasgic & Etruscan barbarians.  La Nature aime les croisements.

      --Ralph Waldo Emerson (Journals)
 

[The] American republic is destined to possess the continent of which it bears the name, and to share it, by absorption, with the inhabitants of all the lands of the earth.  America is the crucible in which European, Asiatic, and African nationalities and peculiarities are smelted into unity.

      --Theodore Poesche and Charles Goepp