By 'clue' I mean cryptic clue - though there's a lot to be said for the witty/clever, off-beat definition clues that sometimes appear in US-style puzzles. A good cryptic clue must be properly constructed, of course. By this I mean:
It has a definition and a cryptic element (or double definition/cryptic definition or an & lit construction).
The cryptic element obeys logical rules enabling the solver to work out the solution - e.g. any anagram is properly indicated, it doesn't contain extraneous words unhelpful to the solver, etc.
It should read well and make sense as a sentence/phrase.
Above all, it should - however ingenious, however devious - be fair to the solver.

So, given that all cryptic clues should have those qualities, what is it that makes some clues special? Well, I'm sure most people will appreciate wit, succinctness and aptness/topicality. Beyond that there are likely to be individual differences of opinion, which I'll discuss in terms of the main differences of outlook to be found between the following groups of crossworders:
1) Crossword setters themselves - as a setter myself I 'appraise' other setters' clues of course. I can be impressed by their clever or humorous clues but I know that such flashes of brilliance often come about by luck. What I really like to see is a clue which, though not great necessarily, has clearly taxed to the full the setter 's skills and resources (I would put Fawley's clue for NYMPHET: Nabokov’s refusal to restrict rate of progress for Lolita? into that category). Good clues for words that are difficult to clue or fresh, creative clues for 'old chestnuts' I appreciate too. Certainly, when I, as a setter, manage to produce the goods for a seemingly 'unclueable' word/phrase, I get the biggest kick of all. One example is the awkward 'CFCs' (refrigerant, etc. gases.) for which I managed to get: They don't improve the atmosphere of Chile’s premier football clubs.

2) Clue-writing contestants/judges - I'm not all that keen on clue-writing competitions (nothing to do with the fact that I'm not very good at them, of course). Don't you think there's something unsatisfactory, sterile, about a clue isolated from its companions and its crossword grid? Successful clues often don't seem to be judged on whether, in a real crossword context, they'd present solvers with a satisfying challenge. The judges often talk of 'elegance' in clue construction, which I take to mean the clue not only reads well (indeed, rolls off the tongue) but is deftly and succinctly expressed or aptly captures the essence of the solution. In the end, though, I feel that a competition clue, to be successful, needs to have a 'stand-out' quality that begs for it to be taken out of its box, as it were, for closer study and admiration. To be honest, I find such a clue just a bit too pleased with itself!

Nevertheless, competitions do produce some ingenious clues. What's more they illustrate the many ways it's possible to clue the same word - and even when two cluers have adopted the same approach, there's often a surprising difference in their chosen wording.
Here are a couple of good examples, taken from an  About.com competition for a clue for FRANKENSTEIN:
Clue 1 - Open, with intense trembling, a classic tale of shock and horror. (by Gordon Murray)
Clue 2 - Patchwork monster ran, sent knife reeling. (by Patti Itterman)

3) Solvers - the most important group, bless 'em! No doubt there are solvers who want clues that are safely within their capability, so they can look clever, or obscure solution words requiring the use of computerised word finders and/or reference books. However, I'm sure the majority of cryptics solvers want be stretched mentally by devious clues to words with which they are, on the whole, familiar. They want something that tests their problem-solving ability rather than their knowledge/memory (though that comes into it). If they can be surprised or made to smile along the way, that's all to the better. For that reason, I think solvers are impressed less by an individual brilliant clue than by the overall effect of all the clues in a puzzle. It's on that basis they'll most likely judge the setter.

Of course, all solvers will stop to admire a setter's ingenuity now and again and, more importantly, take delight in a clue that's exceptionally satisfying to solve - a clue that has what I call the Ah! Of course! factor. This sort of clue often takes a while to solve; it can be quite frustrating - the solver might even become convinced there's been an error by the setter or a misprint. But when the penny drops finally, the solution is so obvious the solver kicks him/herself for not seeing it immediately! For me this the best sort of clue of all. As a setter, I'd love to know the secret of writing such a clue but, alas, I don't think it's possible to predict that a clue will have the Ah! effect. There is one celebrated clue, however - I'm ashamed to say I've forgotten by whom - whose author must surely have realised he'd hit the jackpot.
The clue is simply: GSGE (9,4). (In case you want to work it out for yourself, I'll put the solution at the end of this article.) The example clue seems to break virtually all the 'rules' of good cryptic clueing I listed above. It contains neither definition nor cryptic indication (the solver has to supply these) yet I think it's ultimately fair. Its 'breaking of the rules' provokes agreeable surprise and humour - and it undoubtedly has the Ah! factor.

Well, as I warned at the start, I don't think I've come up with any ground-breaking revelations about the essence of a good cryptic clue. I've enjoyed setting down my thoughts, though, and hope those of you who are still with me have found something of interest and perhaps at least one new angle on the subject.