If something symbolizes a quality or an idea, it stands for that idea — it is a visible thing that makes people think of an invisible one. A white flag symbolizes surrender. The Statue of Liberty symbolizes freedom. In a novel, the broken clock might symbolize a family that cannot move on.

The grammar is worth pinning down, because it is where students slip. The thing you can see goes first, the idea goes after: X symbolizes Y. The dove symbolizes peace, not 'peace symbolizes the dove'. It takes a direct object with no preposition — symbolizes freedom, not 'symbolizes for freedom'. And it does not usually take a person as the subject doing it on purpose; a writer does not symbolize, the writer's images do. If you mean the person's action, say 'she used the storm to symbolize his anger' or 'the storm symbolizes his anger'.

There is a weaker, everyday use where symbolize means something closer to 'be a sign of the times' — 'that empty factory symbolizes what happened to the whole town'. Nobody chose the factory as a symbol; it just captures the situation so well that it works like one. This use is very common in journalism.

Though actually — English has three near-neighbours here and they are not interchangeable. Symbolize is about standing for an idea. Represent is broader and can be literal: a small figure on a map represents a church; a lawyer represents a client; neither is symbolizing. Stand for is the plain everyday version and can also mean 'to be an abbreviation of' — 'UN stands for United Nations', where symbolize would be wrong. When in doubt, if the connection is meaning-and-feeling, use symbolize; if it is standing-in-place-of, use represent.

Spelling: symbolize in American English, symbolise is the common British spelling. Taiwan's textbooks and exams mostly follow the American -ize. Both are correct; be consistent inside one piece of writing.