You might see this mentioned in historical novels. Dukes take precedence over marquesses, etc. I understand that, at formal dinners, this governs who enters the dining room first and who gets seated how far from the head of the table. But does it mean anything else? And does it affect you at all if you have no titles, offices or honors? And is there any other country where a system of “precedence” still exists?
The order of precedence is simply a listing of rank: those who are more important outrank those who are less important. The list in the UK is headed by the monarch, who is the “fount” of all honour. Thereafter it goes through the members of the peerage and the various officers of state. It’s still in use on formal occasions. On informal ones it’s generally ignored - or so a family friend, married into a viscount’s family in the UK, tells me.
I’d expect that most countries would have something similar. In Australia the table is headed by the Queen, who is then followed by her various vice-regal representatives (the GG and the state governors). There’s a similar list for the US.
Also in the UK a wife share’s her husband’s precedence (unless she’s entitled to a higher rank on her own) but a husband cannot derive precedence from his wife. Even Prince Phillip needed a special decree from the Queen to avoid being outranked by his own son (& every other Duke).
It is, of course, completely social/etiquette in nature, having little or nothing to do with legal prerogatives (which are defined by law or “common law” custom/precedent amounting to law). You can find tables of precedence in Debrett’s and books of etiquette. (FWIW, archbishops rank with dukes and ordinary “non-arch” bishops with, IIRC, earls. Heirs to titles rank according to their courtesy title; certain orders of knighthood have precedence ahead of some peers; privy councillors rank fairly high [which takes care of ministers of the crown from the House of Commons]; and so on.)
And yes, non-noble persons do fit into the sequence. Baronets outrank knights bachelor, who outrank those with non-knightly honors (Members of the [Order of the] British Empire, for example). These in turn outrank “esquires,” (which include the landed gentry as well as the bar), who outrank the yeomanry (the rest of us).
Most people in the U.K. go their entire life without ever meeting someone who has a title or an honor. Most people in the U.K. who do have a title or an honor are only at formal events where precedence matters a few times a year. So not only do most people in the U.K. not know or care anything about precedence, but even the comparatively small number of people who have titles or honors rarely think about it. Indeed, I suspect that most people with titles or honors only vaguely know what the order of precedence is, and they would have to look it up if they were asked about it.
The notion that precedence is something that matters to most Britons is rather odd. It exists, but it’s almost as little relevant to ordinary British life as knowing how a White House state dinner works is to ordinary American life. If you want to know about modern British life, you’re not going to learn much about it from reading historical novels.
I guess it’s like castles – Yanks are sometimes fascinated by them because we have none here.
Well, we have Boldt Castle.
Canadians are not “Yanks.”
On my promotion to captain, I was informed by a senior officer in my unit that if I attended any state dinners I would be expected to know where my military rank fit in the precedence, and handed a much-photocopied piece of paper that had a (very blurry) State Department seal on it. In my case it was damn near the bottom, but apparently captains are considerd equivalent to doctors and lawyers (“esquires”) in the precedence, and rank just barely above common gentlemen entitled to bear arms. At least according to the list, lieutenants were grouped in with ordinary gentlemen.
Having served among lieutenants, and having been one myself, I found this last bit particularly hard to swallow. 
So Wikipedia has the list for the US and other countries, and I note that diplomats have a place on it. But what if there are other foreigners present? Suppose, for instance, that Queen Elizabeth is visiting the President… I assume that the President, it being his country, still has highest precedence, but would the Queen (as a head of state) be ahead of, say, the Vice President? What if it were a (non-regent) duke or earl visiting; does he have any more precedence than a random Joe Schmo?
And what about guests of honor who wouldn’t otherwise qualify for any sort of precedence? Suppose, for instance, the President’s alma mater won a sports championship, and he invites the head coach and some of the athletes to a banquet to congrate them. The coach and players probably wouldn’t even be “esquires”, which would put them very low indeed on the list, but on the other hand, the event is being held specifically to honor them, so you can’t have all the random government functionaries present sitting down and eating before them.
Nor is Boldt Castle in Canada (admittedly it’s less than a mile of water from the border). Consider this as equivalent to a native of the area of the guy who expects that you, from your location, avoid gators while driving your pontoon boat to work across the 'Glades in the morning. 
We also have Scotty’s Castle and Hearst Castle and Howden Castle and that’s just in California. Perhaps BrainGlutton meant that we don’t have really old castles, which is mostly true but we have some pretty old forts.
Any time you see a reference to Mrs. Bush as “the First Lady” you’re seeing the American system of precedence in use. She’s called that because as the wife of the President, she has precedence over other spouses and is the first to enter the room, etc.