A while back, a friend introduced me to Harper's magazine, a wholly average monthly digest in every respect, with the exception of the very last page, where you would find the jumbles and word-searches in a normal magazine. In this particular publication, however, a new kind of puzzle was to be found, and it ABSOLUTELY BLEW MY MIND. It was a crossword puzzle, but the clues didn't make any sense. I mean, look at this: "Pe(9)rish ground; a wild type of music"? How did that yield "dixieland"? Even after knowing the answer, it took me almost two hours to find the reasoning behind it. The puzzles were called "cryptic crosswords," and having a good vocabulary wasn't near enough to finish one. Every clue in the larger puzzle hides another, tinier puzzle.
For example, "Returned beer fit for a king (5)". Anyone? No? The answer is "Regal". The answer to the puzzle is right there at the end, "fit for a king". But how to know what word it is? There are a couple of five-letter words for that; royal, noble, grand; just off the top of my head. The trick is, the word "Returned" in this context means that the word is spelled backwards (also used are the words "backwards", "reversed", and once I caught "mirrored"). So the final piece of the puzzle is "beer". Another word for the word "beer" is "lager", which, when reversed, gives us "regal". Of course, just when you get the hang of it, I regret to inform you that that is far from the extent of the confusion. In addition to reversed words, there are anagrams, word searching, and a whole mess of other stuff.
For another example, try "Chaperone shredded corset (6)". The trick to this one is to know that "shredded" indicates that one of the words is an anagram. Which one? "Corset" (You can tell because the answer is only six letters). So what's a six-letter word for "chaperone" using the letters C, E, O, R, S and T? It's "escort". Case closed.
Another good one to look out for is anything involving initials. Often times, you'll see the words "Initially", "For starters" and "At first". These are all indicators. Try "Initially amiable person eats monkey (3)". The clue for this one is the word "monkey". The first letters of the remaining words "Amiable Person Eats" are A, P and E, for the answer "ape".
The fourth most commonly used technique is hiding words inside other words, a clue often called a container. "Make a change and put me in last (5)". The clue is probably "make a change", so when the clue states "put 'me' in 'last'", it means to place the word "me" somewhere into the word "last". As far as I can tell, there is no word spelled like that, so find a synonym for "last" and try to work it that way. The word "end" as a replacement for "last" yields the answer "emend". Every single clue is at least that tricky.
So I guess that's it. there are some other tricky clue types, but you can read about them here. You should be able to solve some of the easier ones simply with the four techniques above. If you want to try some of your own, pick up a copy of Harper's (tell 'em I sent you) or you can just go here. They're pretty tough, so you can also try the so-called beginner's puzzle here.
P.S.: If you can figure out the reasoning behind "Pe(9)rish ground; a wild type of music", post it in the comments.
EDIT: I just found this site, where it explains how to solve a cryptic clue. It also offer dozens of examples for practice, giving you two hints before telling you the answer. A+, good sir.