Like humans, horses can have facial expressions but they are not as elaborate as those of humans. Nonetheless, the expressions provide us with information about how a horse is feeling.

Snapping
Young foals can perform a snapping action with their mouths. They open their mouth and drawback its corners, expose their teeth and open and shut their jaws often without the teeth of the upper jaw making contact with the teeth of the lower jaw. If they do make contact there is a clapping noise which leads some horse behaviourists to describe this as “teeth-clapping”. Other behaviourists have described this as “jaw-waving”.
It is a submissive action to say, “Please don’t harm me because I’m a vulnerable young horse”. The young horse performs the action towards an older, large or strange horse that approaches. At the age of three or thereabouts they stop doing it. It’s purpose, as can be seen, is to protect the young horse. It gives an impression of an aggressive action but it is quite the opposite as it stops aggression. Other horses understand this although humans don’t sometimes.
The origins of the behaviour is in mutual grooming. Mutual grooming is a friendly behaviour carried out by nibbling with teeth. So “snapping” is a symbolic performance of mutual grooming.
Jaw held tensely open with teeth exposed
Conversely, when a horse holds its jaw open exposing their teeth it is a “bite-threat”. It’s a warning to another horse of an imminent bite with the intention of seeing the other horse off to avoid genuine conflict.
Tight-lipped mouth
This is something like the above but a less violent signal.
Stiff mouth
This action might be performed because the horse is tense, fearful, anxious or in pain.
Relaxed mouth
The animal is feeling peaceful and relaxed or perhaps tired or even exhausted.
Drooping lower lip
The horse is feeling sleepy.
Flehmen face
Horses, predominantly stallions, perform the ‘flehmen face’ when they are intently and purposefully sniffing the scent of a mare’s urine or any other particularly interesting smell. I have written extensively about this in domestic and wild cats. I have described it as the flehmen response. In cats it’s the same behavior and the reasons for it is the same. Both cat and horse inhale the air carrying the scent into a specialised organ (Jacobson’s organ) in their muzzle which is able to read odours to a very high level of precision.
In horses, the stallion stretches his head forward and curls the upper lip upwards and back exposing the upper teeth. He sniffs the air with great interest. Sometimes females do it so the description “stallion face” is not entirely accurate.
Wrinkled nostrils
Sometimes horses wrinkled their nostrils in disgust as humans do.
Flared nostrils
When horses are in a state of excitement or higher motion they might flare their nostrils as humans do. Because Arab horses benefit from a “desert-breathing specialisation” their nostrils are always flared. They are not necessarily alert and excited. Other horses flare their nostrils when more alert and excited.
Eyes closed
When the eyes of a horse are closed it might be in pain or exhausted. Conversely, when they are wide open the horse might be fearful, anxious and apprehensive. Half closed and the horse is probably peacefully relaxed.
The angry eye
The eyes bulge and turn backwards. This might indicate hostility but it might also indicate that the horse is looking backwards at something of interest which exposes the whites of their eyes.