I have to strongly disagree with this statement by CD (DE):
.. es ist oft die Abkürzung des Mädchennamens der Mutter des Betreffenden:
... z.B.: John F. Kennedy = John Fitzgerald Kennedy
Although is is not uncommon for Americans to give the mother's maiden name as a middle name for a child, it should be noted that John F. Kennedy was the only one of his siblings that had Fitzgerald (his mother's maiden name) as his middle name.
There is no custom of using the initial of one's mother's maiden name as an initial in one's signature unless one received the mother's maiden name as a middle name at birth. Thus in my family, none of the six children had my mother's maiden name as a middle name, and none of us use her maiden name as an initial.
Here is an example based on how my mother (born around 1930) signs her name:
Named at birth Harriet Ann Smith (Ann was not her mother's middle name), she adopted her husbans's last name, Johnson, upon her marriage and signs her name as Harriet A. Johnson.
On the other hand, an example based on my wife (born in the 1950s) would be a woman who uses her maiden name (Catherine Jane Smith) professionally, and signs it as Catherine J. Smith. However, for most other purposes she has dropped her original middle name and uses her original last name as a middle name and her husband's last name. She signs her name as Catherine S. Johnson.
To answer the original question, most Americans receive a first name, a middle name, and a family name at birth. The family name is typically the father's family name. The other two names can be drawn from a wide variety of names (including the mother's maiden name). In order to help distinguish oneself from people with similar names and to lend greater formality to one's signature, Americans have (until recently) signed their first name, middle initial, family name.
I have read, however, that the use of the middle initial is becoming less common among young adults compared to earlier generations.