The semicolon is one of the most confusing punctuation marks, and many people are really unsure what to do with it. Some use it in place of a colon; others use it where a comma would be correct.
In fact, a semicolon is really a period that’s wearing a comma costume. It’s a full stop, but it’s pretending not to be. In French it’s a “point-virgule”: a period-comma. I think I would prefer to call them recess periods; they’re really periods, but they’re like a short recess between classes, rather than the full stop at the end of the day – you have a class on one side and a class on the other, or, in this case, an independent clause on one side and an independent clause on the other.
Semicolons are not like colons, and they’re not like commas either. Commas are multipurpose things, but one of the things you can’t do is use them to join two syntactically independent clauses; that’s called a comma splice. A colon is like a pair of eyes, looking expectantly. What is on one side of a colon depends on what is on the other side in some way – syntactically and/or thematically.
A semicolon, on the other hand, is like a tightrope walker – you can see the head on top and the one leg carefully balancing (the other is directly behind and not visible). For the tightrope walker to stay balanced, what is on either side must have equal weight: they must be either syntactically independent clauses or complex list items (by complex list items I mean things in a list that have internal punctuation: We went to see some movies: I, Claudius; Dawg, the Bounty Hunter; and Unforgiven).
To know whether a clause is syntactically independent, look for a subject and conjugated verb; you need one of each (unless it’s an imperative) on either side of the recess period.
Bad: He likes going to the races; usually on Sunday.
Good: He likes going to the races; he usually goes on Sunday.
Also look for conjunctions, which make it not syntactically independent.
Bad: He likes going to the races; which he does on Sunday.
Good: He likes going to the races, which he does on Sunday.
Unless you’re using the semicolons to separate complex list items, the rule is that it would still be grammatically correct if you replaced the semicolon with a period – because, really, it is a period, and when you whip off the comma mask, it will reveal itself. Like in a Mozart opera.