How do You Use a Semicolon?

Do not, I repeat Do not (many at first attempt this), try and force the use of it, or reconstruct perfectly fine sentences in an effort to include it - it will always come out awkward and make it worse. Simply, as another answer says, it links two sentences, or better, statements. I disagree somewhat, because often it would seem nonsensical to keep them separated by commas; in this case people who don't like to use, or have little experience in, using semicolons tend to just make it a comma.

Semicolons are, as you say, quite precise punctuation, and can usually be made into, at the expense of rhythm and cadence, a full-stop or comma. It usually depends on the style, context and purpose of the writing. Often in very factual and fast-paced prose, there is virtually no use of them at all; you just want the facts, one by one: you're not reading for enjoyment or being persuaded, therefore there is little need for rhythm, etcetera.

Here, don't try and use one for the sake of using one. In longer, more meditative and thoughtful writing, it is sometimes used nigh on every sentence, eg. In Walden (indeed the full title is 'Walden; or, life in the woods' - a very good book, not just for learning how to use a semicolon). I know some who seem to think that this 'factual' mode of writing is merely modern - fast-paced writing for a fast-paced way of life; the semicolon belongs in the literary domain of the 19th century.

I think this is completely untrue - it is vital for expressing a train of thought clearly. The usually sound advice of On Writing Well by William Zinsser is of this opinion and that one should instead 'rely on the period and the dash', but I think it is because he writes mainly nonfiction.