The British crossword or cryptic crossword consists of a square grid (most commonly sized 15 by 15) with 180 degree rotational symmetry, alternate letters checked. It is characterised by the use of convoluted puzzles as clues and unlike the American crossword, word lengths are always indicated in brackets after the clue.

The form of the British crossword owes its chiefest debt to Ximenes, crossword setter for The Observer from 1939 to his death in 1971. He was the first to demand 180 degree rotational symmetry (he thought it ensured that the northwest and southeast corners would be similar in difficulty), and set the basic form and classification of clues, and insisted that every clue should contain a definition. His book Ximenes on the Art of the Crossword explained his ideas on filling a grid and clue setting. Even today, there is a strong emphasis on the Classics which can be traced directly back to Ximenes' influence. Crossword setters can often say if they are Ximenean or non-Ximenean, with a friendly rivalry between the two.

Each newspaper has its own style. The Sun runs a 'Two Speed Crossword' everyday, with one set of cryptic clues and another set of quick clues, both with the same answers. It is very useful for students of the cryptic crossword.

The Times sees its crossword as being a branded feature and the editor is at pains to edit the clues to a uniform standard. A similar situation exists for many local newspapers like Metro, The Echo, and in magazines.

Private Eye runs an obscene crossword as a regular feature, and uses own nicknames for people (e.g. Brenda is the Queen). (See below for URL)

The Guardian probably has the most colourful crossword. The pseudonym of the setter is published with the crossword, and setters are given a pretty free reign. Many crossword solvers recognise that there are some crossword setters they get on with and some they do not. Guardian setters (called 'compilers') include Araucaria, Bunthorne, Enigmatist, Pasquale, Paul and Shed. There are about 45 in total. Of these, Araucaria is most often called king.

Anagram

The anagram is one of the most popular features of the British crossword. There are literally dozens and dozens of words that may indicate an anagram in a clue: 'about', 'makes', 'cooked', 'results' etc. In common, they suggest that something is to be manipulated, or that something may produce another. The simplest anagram indicator is a question mark.

Telling beer makes aggressive (11) BELLIGERENT (Metro 9 Aug 2004)
Explanation: telling beer is an anagram of belligerent; aggressive = belligerent; 'makes' is the anagram indicator.

Herod's new island (6) RHODES (Metro 9 Aug 2004)
Explanation: Rhodes is a Greek island, 'new' is the anagram indicator, Herod's is an anagram of 'Rhodes'.

Regrets misguiding user (4) RUES (The Sun 11 Aug 2004)
Explanation: Regrets = rues; rues = anagram of user; 'misguiding' is the anagram indicator.

Double literal

Also known as a 'double definition', two definitions are put side-by-side. The aim of the setter is to produce a clue that sounds like something other than is suggested by its parts.

All-round form of entertainment (4) BALL (The Sunday Times 8 Aug 2004)
Explanation: definition 1 = 'all-round form'; definition 2 = 'form of entertainment'.

Potty train (4) LOCO
Explanation: potty = loco (as in 'mad' or 'insane'); train = loco (as in 'locomotive').

Poet's injuries (5) BURNS (The Sun 11 Aug 2004)
Explanation: Poet = Burns (Robert Burns); injuries = burns.

Found

Also called the 'Hidden' clue; the letters of the answer are to be found, in order, in the clue itself. This sort of clue is suggested by words such as 'part of' and 'some'. Also considered an easy clue, so usually no more than one per puzzle.

Point some clue 'as taken (4) EAST (Paul The Guardian 7 Aug 2004)
Explanation: East = point of the compass. cluE 'AS Taken

Port, or something of holiday resort (3) AYR (The Times 13 Aug 2004)
Explanation: Ayr is a port in Scotland; holidAY Resort.

Defame most of the islanders (7) SLANDER (Metro 12 Aug 2004)
Explanation: defame = slander; iSLANDERs.

Sounds like

Every crossword will contain one or two clues that are puns. These are marked as 'we hear', 'hearing', 'sound', or a similar word. Considered an easy type of clue and therefore not often used.

Thankful sound for the road repair (3) TAR (Metro 9 Aug 2004)
Explanation: 'ta' is an English slang term for 'thank you' and sounds like 'tar'; tar is used 'for the road repair'.

Talked of money received by corrupt office-holder (9) INCUMBENT (The Times 13 Aug 2004)
Explanation: money received is 'income', which sounds like 'incum'. Bent = corrupt. Incum-bent. This is a charades type clue, but has a 'sounds like' component.

Quotation

Pooh-poohed by Ximenes, it is retained by some newspapers (in particular The Times and Daily Telegraph) as an occasional feature. It is nothing more than fill-in the blanks in a literary quotation.

Charades

The charade clue is named for the parlour game. This is the most common clue type: the definition occurs at the beginning or end of the clue; the rest of the clue gives an indication of the component parts of the answer. Charade clues very often use abbreviations (best to confer with the list at the back of a good dictionary) and combinations of all the other types of clues.

Shocked maiden burst out when hugged by Donald (10) DUMBSTRUCK (Pasquale The Guardian 7 Aug 2004)
Explanation: shocked = dumbstruck. Maiden = m (cricketing usage); burst out = anagram of burst; Donald = duck. Hugged by = put the letter of duck around the other letters of the answer. Du-m-bstru-ck.

Logic concerning a boy (6) REASON (Metro 12 Aug 2004)
Explanation: logic = reason; concerning = re; a boy = a son.

Education starts with nothing dividing the basics, which is wrong (5) ERROR (Gordius Guardian 16 Mar 2005)
Explanation:
Education starts = the word starts with E;
nothing = 0;
the basics = RRR (the 3 R's: reading, writing and arithmetic);
wrong = E-RR-O-R

Cryptic

This is the heart of the crossword. In this type of clue, the entire clue is the definition. The clue usually takes the form of an extended description that may be convoluted or even deliberately deceptive. Clues are often marked suggestively with a question mark or exclaimation mark.

Litter at a match (8) CONFETTI (Enigmatist The Guardian May 2004)
Explanation: a wedding is a type of match. To an English mind, the first picture that comes to mind is drink cans and paper in a football stadium after a game.

Character for whom dinner repeated eight times? (6) BATMAN (Paul The Guardian 7 Aug 2004)
Explanation: Think of the old Batman theme song.

Or famously:
         ! (3,3,3,1,4) HAS NOT GOT A CLUE (Daily Telegraph)
Explanation: none required.

Ideals

The ideal clue is terse and should be grammatically correct. The ideal clue should read like normal English and not sound forced.

Araucaria gives this example:
I say nothing (3) EGO
Explanation: This is a clue of the charades type. I = ego; say = e.g.; nothing = 0.

Bunthorne treasures a postcard from Araucaria, congratulating him on this clue:
Amundsen's forwarding address? (4) MUSH
Explanation: This is a clue of the cryptic type. Roald Amundsen used huskies to reach the South Pole. Bunthorne assumes that he would have addressed his dogs with the command, 'mush', to make them go forward.

Thanks to SEoD, and Albert Herring for their comments. spiregrain pointed me in the direction of the Private Eye crossword

Further reading:
D. S. Macnutt Ximenes on the Art of the Crossword Methuen 1961 (re-issued by Swallowtail Books 2001)
Sandy Balfour Pretty Girl in Crimson Rose (8) Jeremy P. Tarcher, 2003.
Araucaria Monkey Puzzles: The Ultimate Cryptic Crossword Collection Guardian Books, 2002.

Helpful sites: http://home.freeuk.net/dharrison/ximenes/
http://www.crossword-puzzles.co.uk/
http://www.guardian.co.uk/crossword/subscribe (£25 yearly subscription at time of writing)
http://www.private-eye.co.uk/content/showitem.cfm/issue.1112/section.xword
http://www.guardian.co.uk/weekend/story/0,3605,880719,00.html (The story behind the most cunning crossword clue in history.)