The Speaker
The speaker, the "Mad Girl" of the title, struggles with two simultaneous and mutually reinforcing problems. She longs for the return of a lover, who charmed and seduced her before disappearing. At the same time, she feels that she is losing her sense of reality, and is in fact unsure whether the lover is real or whether she has invented them. Indeed, it is unclear throughout the poem whether the speaker's heartbreak has led to her madness, or whether her madness, prompting her to invent a fickle lover, has caused her heartbreak. She reflects that "I shut my eyes and all the world drops dead," describing the way that she can bring herself temporary relief. But ultimately, the world—which, for her, appears feverishly chaotic and confusing—comes roaring back, leaving her feeling even more destabilized than before. Moreover, when she is able to dispel the overwhelming (and likely imaginary) images around her, she is left undistracted from her loneliness and rejection, leaving her worse off than before.
The Lover
The speaker, unsure whether her lover is even real, describes this person through a veil of mystery and emotional intensity. Not only that, but because, as the speaker points out, she barely even remembers her lover's name, we don't get a lot of specific information about them. Instead, they appear in brief, dreamlike flashes. They're described as having almost magical powers, which they use to persuade and seduce the speaker. However, it's ultimately not clear how much power this lover has. On the one hand, they seem to have made the speaker fall in love with them and then disappeared without a trace, driving her insane without any regard for the consequences. On the other hand, they might be a figment of the speaker's imagination, meaning that they don't actually have any agency at all.