To argue by definition (X is/is not Y) is to argue about the nature of something or someone.  Some definitions are not arguments: "A whale is not a fish" or "A whale is a mammal"; but controversial  definition claims are arguments: "Beauty contests are sexist events," "Euthanasia is not murder," "Car racing is not a sport," and so on.  Definition arguments are sometimes called "criteria-match arguments."  In backing definitional claims, the writer/arguer must establish the criteria for the larger class of things, the "X" term, to which the particular, the "Y" term, allegedly belongs.  The writer of a definition essay would need to devote some space to carefully explaining and establishing that "sexist events foster sexual stereotypes," that "murder is the crime of killing another human being with malice," or whatever the "Y" term may be.  Central to the definition argument, the writer applies the criteria of the "Y" term to match it with the "X" term. 

For example, a large part of a definition argument (essay) may be spent arguing that "Beauty pageants are sexist events because they foster sexual stereotypes."  The writer may do this by case examples and anecdotes, narratives, personal experiences (if applicable and encouraged by the assignment), extended descriptions of typical beauty pageants, outside authorities, and so on.  The writer should consider using any evidence available, discovered through research, note taking, and writing.  After some brainstorming, I might come up with the following list:
    Why else is there a swimsuit competition?  (Two-piece suits are being allowed in some pageants.)
    Evening wear competition suggests "a woman's place is in the bedroom."
    Lame-brained, clichéd questions for contestants.
    Constant smiling (and pearly white teeth) required.
    There are no beauty pageants for men.
Each of these items may lead to several sentences and, perhaps, a paragraph or more.  To show that I have a balanced, sophisticated point of view, I may also discuss possible rebuttals to my argument:        
    Some contestants do win more on the basis of personality and talent than on looks.
    The swimsuits (even two-pieces) are relatively modest.
    Contestants are judged on overall qualities--of which figure and looks are only a part.
    Wet T-shirt contests are sexist, not beauty pageants.
    Men's bodybuilding tournaments are almost completely a matter of physique.
A consideration of opposing views may force the writer to alter the initial view that all beauty pageants are sexist.  The writer may also grant that idea that beauty pageants are in some ways less sexist than other events.  Still, these qualifications do not scuttle the arguer's claim.

As in many essays, definition and other arguments work best when they have an interesting opener and beginning section that piques the reader's interest in the subject matter, a convincing body of proof, and a conclusion that may sum up the essay's major point(s).  Students are advised to also present a strong, focused thesis statement, perhaps near the beginning of the essay, around which the entirety of the essay is focused.