If you work as a contractor/self employed, you have to fill in a self assessment form. I personally don't need to do one, but my partner did, and it took us about 30 minutes last year.
It really doesn't need to be this complex.
I imagine that they will also tell you if you need pay more for the same reason, but that's never happened to me.
I took a job with an employer in London who didn't get my taxes right, but I was none the wiser of course. A few months in, my tax increased by a significant amount from one paycheck to the next, and this tipped me off to something being wrong. I talked with my employer who looked in to it, but this kept on being the same for about three months. At this point, I'd been on the job for maybe 7 months or so. Finally I'm told I have to sort this out with HMRC myself, so I called them. They were very helpful and pleasant to deal with, and the person on the other end of the phone call very briefly looked over my account and determined I've been overpaying every single month – by a lot. They noted as well that HMRC had indeed changed my tax code to an "emergency rate" or some such about three months prior to the call, but failed to note the wrong tax code that was already there in the first place. This person did sort it all out however, and a couple of weeks later I got a check in the mail for over £6000.
It did finally work out in the end, but I'm not sure I would've noticed if they hadn't hiked my tax so significantly, and I don't know what would've happened then since apparently they hadn't registered the error either.
Moral of the story: HMRC are there to help, but you may want to be proactive and call them if you think anything might be wrong, or you want to double check something. Their phone service is very helpful, and usually not too busy.
For most employed people, this calculator should be very close to correct: https://listentotaxman.com/
Years later, I moved to the US. I'm sure it's probably less bad if you've grown up with this system, but I still have no idea what to change (or by how much) to make sure I don't owe taxes next year. I don't understand how to play the tax game, so I'm pretty sure I'm paying much more than I should.
However, a friend has an interestingly different approach: pay as little as possible in taxes throughout the year, set aside what you estimate your tax burden will be, keep the interest, and pay it all in one lump sum at tax time. They refer to this as "not giving the government an interest-free loan".
Edited to add: I never thought I'd say this, but... I miss HMRC.