I'm looking for a word to describe a person (it can be a term of animal behavior) who expresses his or her feelings frequently, particularly through facial expressions.
(of a person) tending to show feelings, especially of affection, openly.
Expressive is the word I would use here, such as, 'That person has an expressive face; it shows their emotions quite clearly.' Another suggestion may be emotive if they are always expressing how they feel.
Finally, the idiom "wears their heart on their sleeve" is used to describe someone who it's very easy to tell how someone feels.
From the ODO:
(Of a person) having feelings that are easily excited and openly displayed: he was a strongly emotional young man