The tympanic membrane is the connective tissue that transmits sound to the middle ear and separates it from the outer ear.
In Human anatomy the tympanic membrane is also known as the eardrum. As sound waves enter the external auditory meatus, the tympanic membrane vibrates which causes the attached ossicular chain in the middle ear to vibrate and ultimately the cochlea in the inner ear to vibrate. Those vibrations stimulate hair cells in the cochlea which send an electrical signal up the auditory nerve to the cortex.
In Ferengi anatomy the tympanic membrane is believed to perform the same function as its Human counterpart. When it gets infected and left untreated for more than three weeks, Ferengi can die from it. A massive infection of the tympanic membrane is a chronic condition among Ferengi. When Grand Nagus Zek faked his death in 2369, the reason stated for his death was that he had this chronic infection of the tympanic membrane. (DS9: "The Nagus")
The infection itself is painful for a Ferengi and can cause dizziness. It can even cause a Ferengi to faint when treatment is postponed too long. Some Ferengi have household remedies to treat the infection themselves. Rom tried one of his mother Ishka's remedies, which involved pouring it into his ear, but it did not work. Medical attention is needed to cure it completely. (DS9: "Bar Association")
For Ferengi, the tympanic membrane can also be stimulated with the help of a feather. This is known as a tympanic tickle. When performed by an amateur the feather can get stuck and medical attention is needed to remove it. Antibiotics are needed to prevent infection of the membrane.