In the opening scene, Egeus goes before Duke Theseus to petition him to grant "the ancient privilege of Athens" (I.i.42). According to ancient law, if a daughter disobeyed her father in any way, the father had the right to either kill her or send her to a convent. Therefore, Egeus is asking Theseus for the right to either execute Hermia or send her to a convent should she continue to refuse to marry Demetrius.
However, it is interesting to note that while sending her to a convent is an option, Egeus actually asks permission to "dispose" of Hermia, either through marriage to Demetrius or through her death. In other words, Egeus has decided to ignore the more humane half of the ancient law and to ask for her to be put to death instead, as we see in his lines:
As she is mine, I may dispose of her;
Which shall be either to this gentleman
Or to her death, according to our law. (43-45)
Regardless, Theseus softens the decree, and when Hermia asks what would happen to her if she still refuses to marry Demetrius, Theseus still includes the option of sending her to a convent in the potential decree, as we see in his lines, "Either to die the death, or to adjure / For ever the society of men" (67-68). This shows us that even though Egeus is demanding Hermia's death, as the ruler, he may decide to take the more humane approach and decide to have her put in a convent instead, should she continue to disobey her father.
Hence, we see that Hermia's fate, if she refuses to marry Demetrius, will either be early death, or life in a convent.
By the Athenian law, if she defies her father's wishes to marry Demetrius, she must be put to death (harsh, right?)
However, Theseus, the Duke, gave her a another option: to become a nun.
Theseus gave Hermia 4 days to decide whether she would marry Demetrius, die, or become a nun. This is also the day of his wedding to Hippolyta, the former Amazonian queen.