Undergraduate: a student in a university or college who
has not received a first, esp. a bachelor's, degree. (
source)
Graduate: a student who holds the bachelor's or the first professional degree and is studying for an advanced degree. (
source)
In other words, undergraduates don't have the more advanced degrees yet, while a graduate has achieved them. A freshman in college is sometimes an undergraduate because it's their first time going in to get one.
Major: To pursue academic studies in a major:
majoring in mathematics. (
source)
The major is what your Undergraduate/Graduate degrees
are for - aka: what you are certified to do.
I'm an undergraduate majoring in the English language, meaning I'm trying to get an Associates or Bachelors degree in the field of English so that I can teach it to others.
Does that answer your question?
eek 
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