Metaphor, Simile, Analogy

A metaphor is often poetically or figuratively (NOT literally) saying something is something else. It creates vivid imagery. The comparisons do not contain "like" or "as" and are directly stated. Examples:

• Life is a highway.
• My life is an open book.
• He is a shining star.
• She is an early bird.
• It was raining cats and dogs.
• You have a heart of gold.
• Fit as a fiddle.
• Dull as dishwater.
• Time is money.

A simile describes one thing by comparing it to another using the words like or as. It is more simple and basic than an analogy. Example - 

• Life is like a box of chocolates
• Apollo is as cool as an ice cube
• Our wedding was like a fairytale

An analogy is saying something is like something else to make some sort of an explanatory point. It explains an idea. Analogy creates a relationship based on parallels or connections between two ideas. It is more elaborate and extensive than a similie.

• A good speech should be like a woman's skirt; long enough to cover the subject and short enough to create interest. 
• Life is like a box of chocolates, you never know what you're going to get 

Analogy uses literary devices like similes and metaphors. 

Similes and metaphors add color and depth to language. 

A metaphor is an implicit simile, while analogy is an explicit one. Put differently, a metaphor is literally false, while an analogy is literally true. 

Metaphors need a bit more imagination to interpret, while analogies are readily apparent.

Source: Basic Instructions

Also see: Difference between Stalemate, Impasse and Mexican Standoff