But more to the point: crossword (puzzle). This word/expression can refer to the same thing in BrE and AmE, but it usually doesn't.
In the UK, the bare term crossword most usually refers to cryptic crossword puzzles. These exist in the US, but not as much as in the UK, where each of the (mainly BrE) broadsheet newspapers has a daily cryptic crossword. Now, these were not the original type of crossword puzzles, and everyone here knows they are cryptic crosswords, but if we look at the adjectives that come before crossword in the British National Corpus, cryptic crossword only occurs once in 100 million words. The most frequent adjective before crossword in the BNC is quick, which names the other kind of crossword that's found in the UK. The reason why quick crossword occurs more than cryptic crossword is not because people write about cryptic crosswords less. It's because when they do write about them, they tend to just say crossword. (Take for example, the Guardian's Crossword Blog, cited again below, which pretty much only discusses the cryptic sort.)
In the US, the word crossword tends to refer to a different animal than is seen in the UK. If one were to talk about those ones in the UK, they'd have to be called American-style crosswords or something like that. If a puzzle is a cryptic one, Americans will call it a cryptic crossword or sometimes a British-style crossword puzzle. Among those in the know, though, British-style crossword refers to a grid style, as opposed to American-style grids. This picture comes from an eHow page on how to make crossword grids. The one on the left, with less white space is British-style. The one on the right is American-style.
Both cryptic crosswords and quick crosswords in the UK are in the British-style grid. (In the US, you might see British-style grids in school exercises, but not usually in newspapers.) The British-style grid means that you pretty much need to be able to determine the answer for every clue. If you don't know the answer for one of the across items in the leftmost puzzle above, knowing all the other answers will get you just a small proportion of the letters in the one you don't know. If all you've figured out is that they want a five-letter musical instrument whose second letter is I, you won't know until the answer is published whether it's a PIANO or a VIOLA (or some other instrument I haven't thought of).
In the American-style one, you can get the answer in a roundabout way. Since each of the letters of the five-letter musical instrument intersects with another word, you can build the word one letter at a time from other clues. But because of this, American clues are much more ambiguous than British ones. For instance, the clues in British crosswords of both types tell you how many letters are in the answer, and how the string breaks down into words. American ones don't give you that information, though the easiest ones might tell you that the answer has multiple words. American clues are sometime jokey (more so than UK Quick ones) and the puzzle itself often has a running theme (so can the other types, but this is a [mostly AmE in this figurative sense] calling card of American puzzles). Because there are so many short words in an American-style puzzle (and they need to line up nicely), any American puzzle-solver has a good vocabulary of three-letter combinations that somehow mean something--including compass points and acronyms.
A quick tour of clues--which won't do any of the puzzles justice:
Cryptic (The Independent Cryptic Crossword 7768 by ANAX as discussed in the Guardian Crossword blog):
26ac What can you get for 20p? Oddly, silver key (4)
The Guardian blogger saw fit to explain the I, but have you got the SLE? Oddly is the clue to tell you to look for--the odd-numbered letters in the following word, silver. (The keyThe answer is ISLE (as in the Florida Keys); the first bit of the wordplay is a plug for the Independent's sister paper i, which belatedly started including a cryptic crossword - one that's as good as any broadsheet's and which we'll look at here in more detail before long.
UK Quick (from Guardian quick crossword 13353):
16 Be transferred by contact or association (3,3) (RUB OFF)
American-style (New York Times, via Rex Parker's blog):
42D: What the Beatles never did (REUNITE)
44A: 1970 hit by Sugarloaf (GREEN-EYED LADY)The last of these was part of a theme (left for the solver to discover) of songs with eye colo(u)rs in their titles.
I love to do the New York Times crossword whenever I get the chance (which isn't much, because when I visit the US all the crossword puzzles in the newspaper are spoken for, and you do NOT do someone ELSE's crossword puzzle. Not if you know what's good for you).*
But I am a fan of the British cryptics--by which I mean that I admire them and like to read about them, but I don't do them myself. (Whenever I convince myself I've got the patience for the clues, I become undone by the inclusion of bits of British cultural knowledge that I don't have--such as anything to do with cricket.) I'm not sure if anyone else sees crossword puzzles as a spectator sport, but it's a good one. And so when Araucaria's cancer puzzle came to light, I was saddened and appreciative [that he wanted to communicate with his fans in this way] as a long-time spectator-fan. As far as I can tell (there's not a lot of data in the corpora), the term crossword compiler is used in both US and UK, but perhaps more in the US, since in the UK crossword setter seems more common. (Recall our discussion of exam-setting too.)
Finally if, like me, you're an crossword-spectating expatriate in the UK with South African connections, I recommend Pretty Girl in Crimson Rose , South-African-expat-in-UK Sandy Balfour's memoir of falling in love over puzzles. You might even like it if you're just some of those things.
*Yes, yes, I could download an app or something. But have I mentioned that I have a job with DEADLINES?! In my life crossword puzzles are for (BrE) holidays/(AmE) vacations or hospital stays. And now that my holidays/vacations involve a child, they're not really for those either.
P.S. (the next day): @MagdalenB sent me this on Twitter. A British crossword setter explains the differences between British and American crosswords (after 2 long minutes of introduction, which can be skipped). I'm right about the cricket!!