The Eustachian tubes are tubes that link the middle ear to the nasopharynx. They serve to regulate the pressure in the middle ears, allowing air in and draining mucus out.
Your middle ear is home to the tiny ossicles, AKA the malleus, incus, and the stapes. These bones conduct sound best when the middle ear is filled with air at normal pressure (or something close to it). The eardrum will also vibrate most effectively when pressure is equal on both sides. Pressure in the middle ear may become lowered when the mucus membranes absorb oxygen, taking it up into the bloodstream. If the Eustachian tubes are clogged up this pressure differential can become painful. Changes in external atmospheric pressure, as when driving up a mountain or flying in an airplane, can also become painful, but a hard swallow usually opens the tubes enough to resolve the pressure differential.
The middle ear also produces mucus, particularly if it becomes infected. In the general course of things this mucus should drain out into the nasopharynx, but if the mucus becomes too thick to drain (due to infection) or if the Eustachian tubes become inflamed, the mucus may build up, causing increased pressure in the middle ear, which will in turn cause painful ear infections. In this case the pressure often only reaches equilibrium after the infection is successfully fought off or the eardrum bursts (or is pierced surgically). While the pressure behind the eardrum is extremely painful, it is not a bad thing to have the drum burst; it will repair itself fully. However, chronic infections that cannot drain through the Eustachian tubes or the eardrum can cause serious (even deadly) long-term infection, and you should see a doctor about any chronic infection.
As any parent knows, ear infections are more common in children than in adults. This is in part because the Eustachian tubes start out level, and only develop a downwards slant (towards the nasopharynx) to aid in drainage around the age of four to six years. This allows material, including bacteria, to move with comparative ease 'up' the Eustachian tube to the middle ear. This is one reason that it is recommended that you feed babies and infants with their heads elevated; milk entering the nasopharynx encourages bacterial growth and infection that may spread up the tubes to the ears. Children also have narrower tubes, making it harder to drain mucus, and shorter tubes, meaning that mouth-born bacteria have less distance to travel to reach the middle ear.
In adults the Eustachian tube is about 35mm (1.5 inches) long, and has a clear slant down from the ears to the nasopharynx. It opens from the bottom of the anterior wall of the middle ear and exits from the lateral wall of the nasopharynx. When the tube exits the middle ear the first 1/3 of the tube travels through bone lined with mucus membrane; this is referred to as the osseous portion of the Eustachian tube, and as you might expect, this portion always remains open. As it approaches the nasopharynx the other 2/3 of the tube becomes cartilage; the cartilaginous portion of the Eustachian tube usually remains closed, and only opens when dilated by the muscles of the nasopharynx, most notably the tensor veli palatini. The salpingopharyngeus, levator veli palatini, and tensor tympani may also help open the tube. It can also be forced open by strong pressure differentials; if you have an ear that just wont pop, you may be able to pop it by pinching your nose closed, closing your mouth, and trying to exhale strongly up your nose.
The Eustachian tubes can become inflamed or swollen due to infection or allergies; this can be painful and annoying on its own, but it can also damage the health of you middle ear, as it can no longer drain. Allergy medicines and antibiotics may be called for, although getting a general practitioner, or even a ENT specialist, to treat Eustachian tube problems can be difficult.
The Eustachian tubes are named after Bartolomeo Eustachi, the man who discovered them, and as such should always be capitalized.