Metaphor is implied comparison between 2 objects that are basically different, but in reality have something in common. It has: A -the tenor, B - the vehicle.
Formula: A is B. Here we don't use like or as.
Simile we can find with the help of linking words: like, as, to resemble, to appear and others. It is the imaginative comparison of 2 unlike objects that belong to different classes.
Formula: A is like B.
Personification is giving attributing qualities, abilities or characteristic features of human-being to animals or lifeless objects, making abstract objects behave as alive.
- Personification creates a visual imagery in a manner so beautiful, that you are tempted to imagine what you read.
- The use of this literary device gives a break from the monotonous language that you may be used to reading all the time.
- By giving human qualities to an object, personification serves to emphasize that object, to give the object its own identity, and explain its place in the piece of literature.
- The use of personification also enables the object, action, or concept that is personified, to communicate with the reader.(http://www.buzzle.com/articles/what-is-personification.html)
Metonymy
Metonymy is applying the name of an object to another object that is in some way connected with the first.
For example: The gallery applauded. - We imply the people in the gallery, NOT the building of it. (Yu. M. Skrebnev)
Metonymy has some subtypes of it. We learned about synecdoche and antonomasia.
Synecdoche is the use name of a part to denote the whole or vice versa.
Antonomasia is a substitution of any epithet or phrase for a proper name.(http://en.academic.ru/dic.nsf/enwiki/373130)
Simile. Differences and similarities between similes and metaphors.
Simile is imaginative comparison of a partial affinity of 2 objects. It is used with the help of link words:like and as (also: to resemble, to seem, to look like).
As well as a metaphor, it has the tenor and the vehicle.
E.g. The rainbow - is the tenor and its descriptions are vehicles (words after as)
Both similes and metaphors compare things, but they do it in a different way.
We can see their formulas: 
As we see on the picture metaphor makes more direct comparison of two objects.
In everyday life we use them and we don't give attention to it, but without metaphors and similes our favourite songs and may be our life would become boring. You can check it here http://www.youtube.com/watch?v=zNFeAgUzpEQ
Metaphor
A metaphor is a figure of speech in which there is an indirect comparison or contrast between some words, without using 'like' or 'as'. These words always have something in common in their meaning.
A metaphor expresses unfamiliar (the tenor) in terms of the familiar (the vehicle). The tenor is equal to the vehicle, formula: [A is B].
Metaphors can be divided in two ways - semantically and structurally.
Semantically: original (E.g .It's been a purple dinosaur of a day) and trite (E.g. A blanket of snow covered the garden).
Structurally: simple (E.g. The UK is a melting pot of cultures), prolonged (E.g. It will take a big tractor to plow the fertile fields of his mind. ), mixed (E.g. It was playing with fire in the belly.)
"Metaphor is a device for seeing something in terms of something else. It brings out the thisness of a that, or the thatness of a this." (Kenneth Burke, A Grammar of Motives, 1945)
An interesting video about our topic: http://www.youtube.com/watch?v=eFgRj-5d5Ac&feature=player_detailpage