I am uncertain as to their usage, this after reading many descriptions and examples, including in teacher training books. I gave up a long time ago and mostly avoid them but occasionally stick them in longer sentences which I feel don't need to be broken up.
Can the august posters on the forum please illuminate me?
Multiple, but two predominate.
There are some differences in accepted practice between British and U.S. English.
The less-common reason is to serve as an additional form of comma in lists where commas have already been employed. The semicolons serve to 'group' the items.
You will need to bring the following: sleeping bag, pillow, and pyjamas for the overnight stay; water bottle, waterproof jacket, sweatshirt, and walking boots for the afternoon trek; and a swimming kit for the river activities. [Stolen example as I'm feeling unimaginative]
The more important use is to join independent clauses (a full sentence or a part of a longer sentence that
could stand on its own grammatically). These clauses are either linked grammatically, causally or thematically (or in multiple ways--see below), and the 'linking' often allows you to omit the kind of conjunctions that would normally be employed to specify the nature of that link.
Steve had a rocket launcher on his shoulder, a rifle slung across his back and enough ammunition to sink a battleship's big brother; no fucker stood a chance once he walked into the room.
In this case the 'link' is twofold: there's a grammatical link where 'he' in the second clause refers back to 'Steve' in the first, and a causal link where the second clause is a consequence of the first (and there's no need for 'because' or 'hence' etc.). The idea is it's not a substitute for an equals sign like a colon is (The greatest ever team: Tottenham), but not as vague as a comma.
Very often the clause after the semi-colon sums up or adds commentary/judgment on the clause that preceded it. One distinction between U.K./U.S. usage is whether the semi-colon
must be followed by an independent clause (minimum: subject and predicate, i.e. noun and verb). You sometimes see U.K. constructions where they're a pithy addition, not a full clause.
Came home to find my wife had just got back with my favourite fish and chips; fucking magic.
Personally, I'd use a dash there.
I could write more, but see whether that's clear first.
The third big function is joining clauses with conjunctive adverbs.