Based on Archayanami's Female Only Challenge for Fire Emblem 7 (Blazing Sword), the Fire Emblem Awakening female only challenge basically is what it says on the label: you are not allowed to use any male characters, except for Chrom (because he is a required character). The player's avatar must also be female. Surprisingly, most versions of this challenge run allow for unequipped males to be used to gain supports with a female character, and furthermore, marriage, if that female has a future female child. After that, the male may not be used.
There are a few other self imposed rules that may go with this challenge, including no DLC (characters or levels), using Classic Mode and a higher difficulty, a limited number of resets, and no marriages except for Chrom.
This can also be done the other way around with only male characters.