A semi-colon is used to draw two completely formed and independent sentences together more closely. You can easily replace the semi-colon with a full stop/period, and not lose any meaning at all; you may just lose the very close relation of the two sentences (as I've just illustrated in this sentence). The most common error here is using a comma instead of a semi-colon or full stop, this is called a "comma splice" and I've just done it.
A semi-colon is also used as a separator instead of a comma, in a list where there are other necessary commas. "There were several choices on the menu: bacon, eggs and toast; green eggs and ham; apple pie with ice cream; and a selection of English, French and Spanish cheeses.
A semi-colon also separates clauses that begin with however and therefore; therefore, you should use it thus.
A colon precedes a list, but is only used when the introduction is, by itself, a properly formed complete sentence. Go back to my menu example above. You cannot write, "On the menu were: item, item, item." A colon is not required there. Nothing is required after the word were.
A colon also is used after a complete sentence to give you an example: there, I've just done it.