A comparison between two unrelated things; does NOT use the words "like" or "as."
Note to all of the people out there who only think they understand grammar:
A comparison using "like" or "as" (definition #2) is a SIMILE.
A sentence used to express the relationship of words (definition #3) is an ANALOGY.
"America is a melting pot" is a metaphor.
"Life is like a box of chocolates" is a simile.
"Pink is to red as gray is to black" is an analogy.