The semicolon (;) is typically used in two instances in an essay. The first is to separate a list that already uses a comma (,). The second and more important usage is to separate two independent clauses in a compound sentence. Perhaps the best way to explain this is to offer an example.
"The boy ran fast in the football game; he scored the winning touch down."
What we have above is a compound sentence, which is conjoined by a semicolon. It gives the sense of immediacy and creates a close connection between the two sentences.
The colon is used to express an example from the previous independent clause. Here is an example.
"A way to remember which direction to move the hands of the clock when changing to or from Daylight Savings Time: spring forward, fall back."
These examples should give you an working knowledge of the difference between the colon and semicolon.