Well, you've come to the right place. You see, semicolons and I have a long, intricate history together; a history riddled with their overuse.
Semicolons are, first and foremost, for those situations where you kinda-sorta-maybe want to start a new sentence, but the two ideas can still be held into one. AKA, they're wonderful for accidentally causing run-ons. Most of the time, their use is purely optional, and just starting another sentence would be perfectly acceptable; however, there are some times where it makes more sense to hold the sentence together. It's definitely not a good idea to overuse them, though; enough people don't fully understand their meaning that it's not worth confusing your readers by throwing around fancy punctuation.
Quite often - but not always - people use an ellipse (...) where a semicolon might be more appropriate:
I thought about it for a while... then, the answer struck me.
vs
I thought about it for a while; then the answer struck me.
If anything, in my mind at least, it conveys a sort of abruptness; the idea following the semicolon is so importantly connected to the idea before that it
must be in the same sentence, but has enough importance on its own to be emphasized and separated in some form.
Once again, I state: it's quite often regarded as nothing more than a run-on generator. You've got to be very careful when using it, or else you could go on for half a page with a single sentence. A good rule of thumb is never more than one per sentence, or paragraph, for that matter.
The exception is when using a semicolon to separate out a list; that is, when the individual items in the list contain commas of their own, and trying to separate them otherwise would cause confusion. For example:
Sarah brought along her dad, Fred; her mom, Annie; her brother, Doug; and their dog, Sparky.
If there were only commas there, the sentence would flow very awkwardly, and it would take a couple reads to realize that her dad and Fred were the same person, and so on down the list. In this case, semicolons provide a clear dividing line between each item.
For a slightly more professional explanation:
http://en.wikipedia.org/wiki/Semicolon