Good answers so far. I thought I’d just answer your specific questions a bit more.
GCSEs aren’t similar to the SAT at all. They’re mostly essay-based, and you take two or three exams per subject, and usually about 10 subjects, plus there might be some coursework. Here’s an example paper for English: http://filestore.aqa.org.uk/sample-papers-and-mark-schemes/2017/november/AQA-87001-QP-NOV17-CR.PDF
Recent changes took away a lot of the coursework elements except for practical subjects like drama, and the grading was changed from A*-G to 9-1, with 9 being the highest grade.
The only similarity to the SAT is that the marking is done blind, so it doesn’t matter where you studied, an A is an A. Well, if you got an A at a rough inner city school you might well be brighter than a kid at an expensive private school, but the exam itself is the exact same thing. So it doesn’t matter if your teachers want to bump your grade up or down, they can’t.
I think the name is a bit confusing; General Certificate of Secondary Education does sound like it’s one single certificate, like a high school diploma. But it’s always plural. “He took his GCSEs this year.”
And yep, they’re the replacement for O levels.
A levels are the exams you take at 18 (usually), like the others said. You don’t attend additional classes or do self-study, they’re what you’re in school to study for. The level is similar to an AP course or slightly higher. Again they’re marked blind. Some A levels, like drama and art, include coursework, but it’s mostly exams.
AS levels (Advanced Supplementary) are, essentially, in between GCSEs and A levels. It’s fairly common to do an extra AS level as well as three or four A levels (this is also in a state of flux but that’s a level of detail you don’t really need). So you might do physics, chemistry and biology if you want to study medicine, with maths or a language or psychology or almost anything, really, as an AS level as well.
There are alternatives, like BTEC awards, usually for more practical subjects. The IBac is another option but very few schools offer it, and you couldn’t, as a pupil, say “I want to do this instead.”
This is how it works in England and Wales. Scotland’s system is completely different.
You don’t choose a single subject specific A level, you do three or, very often these days, 4. The girl at the college was exaggerating a bit. It definitely helps to choose the right A levels for your specific course, but unless you decide to switch from history to medicine, or something like that, it won’t usually be a problem. Then you’d have to take extra A levels, which you can do at any age.
At university you do specialise early on, but if you realise after the first term/semester (some universities have terms, some have semesters), that you’re on the wrong course it’s usually possible to switch paths.