Irony occurs when words mean the opposite of what they state, when situations are the opposite of what is expected, and when readers know what characters in a work of literature do not.
Verbal irony occurs when words mean the opposite of what they say. In the opening of her essay, Hurston uses verbal irony when she states,
I am the only Negro in the United States whose grandfather on the mother's side was not an Indian chief.
Of course, what she really means is that while many Black people claim to be descended from Indian chiefs, this is not really the case. Hurston is hardly the only Black person not to be of Native American descent. This statement, however, is humorous and helps establish the lighthearted tone of the essay.
Situational irony emerges in the paragraph in which Hurston calls herself the "cosmic Zora" who "belongs to no race or time." This is ironic because earlier in the paragraph, Hurston locates herself solidly in a specific time period, the 1920s, to the point a modern reader probably does not understand some of her references. She compares herself favorably, for instance, to "Peggy Hopkins Joyce" in her "gorgeous raiment." Most readers today have probably never heard of Joyce, once a popular silent-movie star, and would miss the irony in Hurston comparing herself to someone known for her blond hair and pale skin.
Finally, from the distance of almost a hundred years, a dramatic irony emerges in how greater Black equality has been achieved. As we know, but Hurston's narrator could not, it has not come primarily from a positive attitude and individuals doing battle to better themselves, but through concerted social justice efforts.