Well thank you for the website, though I must admit that some of what it was saying went right over my head! For example...
"Use a semicolon... Between independent clauses containing internal punctuation – even when the clauses are joined by coordinating conjunctions."
I enjoy writing, but I admit that my grasp of the actual rules and terms are elementary at best.
In my experience, people are most likely to make mistakes by using a comma when a semicolon should be used. For example, I need a comma in the current sentence. This is because 'for example' is a phrase that makes no sense on its own: it needs to be modifying 'I need a comma in the current sentence'. So, I use a comma to show that there are two units, the first of which depends upon the other.
However, as soon as I want to link two grammatical units that are both sentences in their own right, I must use a semicolon.
Basically, if you have two pieces of text which are complete sentences in their own right (eg, 'I am quite pathetically interested in grammar' and 'Some people think I am a terrible pedant'), these can be joined by a semicolon ('I am quite pathetically interested in grammar; some people think I am a terrible pedant'). This removes the need for any kind of connecting word, like 'and' or 'but'.
A semicolon is used when two grammatical units are of equal 'value', that is, when neither one is modifying the other. In the above example, the two pieces of text could each be a single, stand-along sentence: neither is incomplete without the other.
It would be incorrect to join them together with a comma (this is not a sufficiently 'heavy' punctuation mark to mark the work of each grammatical unit'), so either a conjunction like 'and' or 'but' is used, or a semicolon.