In the U.S. in by far the most common usage, in the name "Norman Z. Moody,"
"Norman" is the "first name,"
"Z." is the middle initial, and
"Moody" is the last name.
In "Norman Zachary Moody,"
"Norman is the first name,"
"Zachary" is the middle name, and
"Moody" is the last name.
The last name is often called the "surname." A person's surname is the same as their father's, by law and custom, unless someone goes through a legal procedure to have it changed.
The first name is sometime called the "given name."
It is now rare and old-fashioned, but you will see given names referred to as "Christian name" or "baptismal name" because in Christian traditions that practice infant baptism, as far as the church is concerned, the child is officially named when it is baptized.
The first and middle names are chosen by the parents.
Not all people in the U.S. have middle names.
Occasionally people have more than one middle name--some Catholics have the two middle initials F X., as in Robert F.X. Sillerman, for Francis Xavier.
Rarely, there can be a middle initial that does not stand for anything. U.S. president Harry S. Truman is a famous example--the "S." is just a letter.
It is not unusual for people to decide that they like their middle names better than their first names. When they do, they may choose to go by their middle name. Thus, a famous writer whose full name was Francis Scott Key Fitzgerald is known as "F. Scott Fitzgerald," and his friends called him "Scott."