Facies refer to a collection of symptoms that can be easily spotted just by looking at someone's face. As the name suggests, in this case the underlying cause of the symptoms is often swollen adenoids, which most of us simply call tonsils.
Chronic swollen adenoids cause a number of visible symptoms. Mouth breathing leads to a slack jaw and forward tongue placement; over the long term this may also cause an under-developed jaw and bite, meaning a shorter upper lip, a poor bite, tooth crowding, prominent upper teeth, and a high palate. Snoring and apnea are often present as well.
When the facies are caused by chronic upper respiratory tract infections (which may or may not be the result of infected adenoids), you are likely to see additional symptoms. A runny nose leads to a nasal pleat. Swelling tissues in the sinuses can cause blood to pool under the eyes, resulting in rings under the eyes, sometimes so severely that they cause allergic shiners which closely resemble black eyes and deep folds in the lower eyelids.
At a quick glace, this looks like an elongated, slack-jaw face with small nostrils and rings under the eyes; it is also common to look sleepy or glass-eyed, as getting less oxygen leads to low attention and engagement.
Adenoid facies is a diagnostic indicator for upper airway obstruction. In children this is usually a problem with the adenoids, but could also be any number of other things, from allergic rhinitis to Cowden syndrome to a coin up the nose. It can also lead to other conditions, such as frequent ear infections and potentially, deafness.