As I understand it, emotion comes from deep within the limbic system. This is an ancient area of the brain being present in reptiles (and birds) as well as mammals. Thus I imagine the basic emotions (e.g. fear, rage, lust) would be shared across all these groups in some form. As the human mind is more developed it probably has a greater range of emotions, as well as the consciousness to feel them. (Also be careful not to anthropomorphize as much as I suspect I have done below).
Here are some ideas of animal emotions I could think of:
The exhilaration of flying (e.g. a crow).
The pain mixed with satisfaction as a snake's body stretches to swallow prey as large as itself.
Dreams during torpor (e.g. hibernation) or the feeling recovering from torpor.
Emotions in animals with senses we lack. E.g. pain/confusion of a platypus in the presence of abnormally high voltages; or a dolphin subjected to abnormal sonar signals).
There are also many human emotions that humans can feel but you may never feel them:
The adoration of a subordinate social animal in the presence of a superior e.g. chimps or baboons. It is said that just reaching out to touch the alpha can cure minor ailments. Humans may experience this with extreme hero worship or religious zeal. Mysterium tremendum (feeling "utterly insignificant but not personally alienated/ absolute astonishment/ awe) in the face of the numinous.
Emotion going through the brain of a being that is dying ("life flashing before your eyes", maybe?)
Extreme stress/fear/exhilaration, as during a high speed chase or gun fight, where your vision tunnels, you go deaf and get temporary amnesia etc.
Terror mixed with boredom e.g. when sitting on a small boat next to a vicious tiger, for long periods of time.
Emotions linked to drug use.
Emotions in someone suffering from a disorder of the emotions e.g. depression or autism.
I am not a neuropsychologist, ethologist, psychoanalyst, artist, druggie, or mind reader.