A diagram of a completed tracheotomy, courtesy Jeremykemp, via Wikipedia
A tracheotomy is a surgical procedure where a hole is cut through the trachea (the fold of skin and cartilage where the neck meets the chest) into the airway to allow the passage of air into the lungs through the opening. It's one of the oldest surgical procedures known, having been depicted in Egyptian artifacts dating back to 3600 B.C.E..
Tracheotomies are performed in emergency situations where swelling to, damage to, or blockage of the throat will not allow passage of air through the mouth to the lungs.
Tracheotomies are also common in patients with cancer of the throat or larynx where the airways are permanently damaged by the surgery used to remove tumors from the throat.
A tracheotomy is performed by Robert Chase in Pilot when Rebecca Adler's airway shuts closed during a MRI.
House performs a tracheotomy on Nick (prisoner) in Twenty Vicodin when he suddenly goes into anaphylactic shock. House uses a pen to mimick a regular pipe.
The doctors perform a tracheotomy on Tommy in Perils of Paranoia and find the blockage is further down the throat.