Hypernasality refers to a resonance disorder in which a person's speech sounds odd because too much air is escaping through the nose.
Generally, when people think of hypernasal voices they think of mild cases like Fran Drescher (who played the starring role in The Nanny) and singers Bob Dylan and Tina Turner. These are very mild cases, and would not generally be considered problematic.
If you are uncertain what mild hypernasality sounds like, children will often try to emulate it by pinching their nose shut. This obviously does not directly cause hypernasality; if anything it would cause hyponasality, which is essentially the sound you get when you have a stuffed up nose. However, in true hyponasality the back of the nose is stuffed up; in 'fake hypernasality', the blockage of air at the front of your nose cues you to open your soft palate as wide as possible, allowing more air into your nasal sinuses. This gives you the high, sharp resonance associated with mild hypernasality, without any of the speech sound problems associated with severe hypernasality.
The technical explanation for hypernasality is that there is a problem with velopharyngeal closure and too much sound energy is escaping into the nasal cavity. Velopharyngeal closure is a fancy way of saying that the soft palate (AKA the velum) is not closing against the back of the throat (the pharynx), leaving the passageway betwixt the mouth and the nose open. This may be due to a shortened or compromised soft palate, nerve or muscle damage, or a problem in muscle planning, such as apraxia. Occasionally children may develop temporary hypernasality after having their adenoid tonsils removed, if the tonsils were helping form part of the seal between the nose and the mouth.
In severe cases hypernasality can be a serious problem, and speech may become 'muffled' or even unintelligible. Because air is escaping through the nose, sounds that require air pressure to build up in the mouth may be distorted. plosives (/p/, /b/, /t/, /d/, /k/, and /g/) and fricatives (/f/, /v/, /s/, /z/, and other sounds made by forcing an airstream through a constricted space) will be hard to make, because the pressure build-up needed to force these sounds past the tongue or lips can't form when all the air can simply escape through the nose.
In children who have severe cleft palate (much the same as a cleft lip, but at the back of the mouth rather than the front), it can be impossible to make these sounds, and they will often learn to substitute a glottal stop in place of these consonants. A glottal stop is essentially an un-sound -- a stopping of the airflow through the throat, leaving a brief silence in the flow of speech. This leaves the speech as essentially a series of interrupted vowels, which is quite hard to understand.
In less severe cases there may be a constant hypernasal quality to the voice or nasal emissions -- sounding like a small gasp of air through the nose -- most frequently occurring on /b/'s and /p/'s.
Severe hypernasality is often treated through surgery to reduce the size of the velopharyngeal opening. In cases where the soft palate is adequate but too weak to close on its own, a prosthetic appliance called a palatal lift may be used. It looks something like a retainer, fitting against the top of the mouth, with a long 'tail' protruding into the back of the mouth, where it will push up against the soft palate. In many cases children will need voice therapy to learn to speak with normal resonance even after the underlying structural problems are repaired.
In the field of phonetics and phonetic transcription hypernasality is referred to as nasalization.