Final answer:
The membrane between the middle and inner ear that the stapes inserts into is the oval window.
Step-by-step explanation:
The membrane between the middle and inner ear that the stapes inserts into is called the oval window. When the stapes presses into the oval window, it causes movement in the fluid inside the cochlea, stimulating the hair cells. This membrane is not the tympanic membrane or eardrum, cochlear membrane, or vestibular membrane.