What is the difference between personification, anthropomorphism and pathetic fallacy?

These three literary devices are often misidentified and confused with one another. However, there are some easy ways to work out which one you are observing in use in a given text.

Personification refers to the attribution of human characteristics to something non-human, or the representation of an abstract concept or quality in human form. Personified objects are not actually portrayed as posessing the human characteristics attributed to them. "The trees danced in the high winds", is a basic example of personification. The trees "danced", but the 'dancing' refers to the motion of trees as they are moved by the wind, not to trees that are literally dancing.

Anthropomorphism also refers to the attribution of human characteristics to something non-human, however it is distinct from personification in that the anthropomorphised non-humans are actually presented as  behaving as though they are human beings. The pigs in George Orwell's Animal Farm are an example of anthropomorphism, as is the Lion in The Wizard of Oz; they are portrayed as speaking and thinking like humans, and posess human intentions.

Pathetic fallacy refers to the projection of human emotions onto surroundings and nature. Emotions associated with occurences in a narrative, or the mood and tone of characters and speakers, are reflected through inanimate objects or the weather. It is effectively a specific type of personification. This is an excellent example from Robert Browning's Porphyria's Lover:

"The sullen wind was soon awake, 
It tore the elm-tops down for spite, 
and did its worst to vex the lake..."

The "sullen" wind, tearing trees down out of "spite", mirrors the feelings and demeanor of Browning's speaker.