I'm English and it definitely exists, but it's not really talked about often. People just generally know which they belong to and - among their peers - sometimes denigrate in both directions.
I suppose there are still technically three classes. It's hard to explain each because there are lots of different factors. Plus - sometimes it's easy to fall into what some would see as judgemental descriptions - hence people keep it 'nice' and miss things out.
Upper - Only really the aristrocracy, so you're born into it and remain part of it. You can marry into it - see Kate Middleton - but will potentially still be judged. Even now some newspapers will refer to Kate as "the daughter of an air hostess", which is another way of saying she'll never truly be one of them.
Middle - Split into:
Upper middle class - ie those who are privately educated and wealthy - they have connections which ensures they keep their status
Middle class or lower middle class - this is people who are in professional jobs/educated/live in 'decent' areas, etc. Quite a few people who were raised working class, but got decent jobs and earnt enough money to buy a nice house when the market was cheaper, successfully moved into this class. Sometimes their accent hints at their roots, but they're accepted as they often have signs of wealth.
Working - Classic blue collar workers. Generally less well educated, poorer, live in dodgy areas, etc. Sometimes hard to get out of given they start with little and can get trapped in unskilled jobs, which leads to a cycle.
You can move between classes but it's not that easy and you'll probably always be a bit judged by some even if you do it. For example, if someone who was raised working class does well at school and manages to get to university and marry someone who's middle class, they might still have (and I hesitate to use this phrase) a "common accent", so other middle class people will potentially look down on them a bit.
But working class people will sometimes mock "posh accents" and prissiness, so it works in both ways really.
It's not like we have committees where we decide the class of everyone, it's just something you can instinctively tell.