Cryptic crosswords are great fun to try, but people are often put off because they just don't know where to start, and because the clues just don't seem to make sense.
I found out most of what I know about solving cryptic crosswords by trying a crossword, and then looking at the solutions, and trying to understand how the answer was arrived at, and this is probably the best way of understanding how it works, but have a read through this and see if it helps.

A brief outline of how most cryptic crossword clues work:
The clue has two parts - a definition and a cryptic part.
The cryptic part gives the letters needed to form the answer, using full words, part words, single letters & 'sounds like's, combining them using things such as anagrams, putting one bit inside another, reversing bits and other ways too, including leaving them as they are.

Here is a more comprehensive explanation:
Pretty much all clues in a cryptic crossword consist of two parts.
Aside: The skill in setting difficult crosswords is making the two parts of the clue blend together into a valid statement, which draws the solver into reading it as a whole rather than as two parts. There are occasionally clues which have just one part, where one must think laterally to solve them, but they are not usually easy to spot, and these are the ones that non-regular crossword solvers tend to get more easily, as you don't need to know any of the 'rules' or 'tricks' of cryptic crosswords to get them.
One of these two parts will almost always be simply a definition of the answer. These are not necessarily dictionary definitions, but are pretty accurate, and sometimes quite clever in themselves.

So essentially solving a cryptic crossword comes down to two things - splitting the clue into the correct two parts, and solving the cryptic part so that it matches the definition part. So how does one go about solving the cryptic part?
It all hinges around the idea of an 'indicator'. The indicator tells the solver how they are to use some or all of the cryptic part, and if no indicator is present, the word or words (or group of letters) just remain the same. Each indicator has a vast variety of words that can be used, some of which are more obvious than others. So what could an indicator indicate?

'Sounds like' - giving clues to words that sound like the answer, or part of it
Words or phrases to indicate this include 'we hear' and 'reportedly'
'Single letter' - indicates individual letters (not always indicators - could just be the letter itself, particularly in the case of 'a').
A lot of these are quite clever (eg 'egghead' = e, 'dead end' = d), others are just accepted abbreviations (eg 'time' = t, '50' = L). There are all sorts of cunning ways of indicating a letter, and you have to be permanently on the lookout for them. You will also often see 'initially' meaning the first letter of a word.
'Letter groups' - indicates groups of usually 2-3 letters.
These are sometimes simply acronyms (eg 'Prime Minister' = PM), but tend to be slightly disguised (eg 'queen' = ER). Often you will discover these by accident because you've already worked out the answer from the definition and from the rest of the cryptic part, and you then realise how the two letters left over were represented in the clue! There are some particularly popular ones which you'll pick up after your first dozen or so crosswords. Words like 'half' or 'part' might indicate that you need to use some of a whole word.
'Anagram' - indicates that you have to rearrange some given letters to form all or part of the answer
Indicators used here range from the obvious (eg 'rearranged', 'organised') to the more obscure (eg 'broken', 'again'). There are literally hundreds of anagram indicators, you will pick up the more common ones as you do more crosswords, but what I often do is count the number of letters in words in the clue, and see if I can combine them to make the right number for the answer, and only if I can do that look to see if there could be an anagram indicator.
(If you are interested in anagrams more generally, have a look at Anagram Genius.)
'Hidden' - indicates that the answer is hidden inside the rest of the clue, as a consecutive sequence of letters, possibly over more than one word
Usually quite obvious (eg 'inside'), although sometimes just 'of' or 'in' are used to indicate that the answer is part of a word, or words.
'Container' - indicates that one sequence of letters must be put consecutively inside another to from all or part of the answer (can be done either way - 'A' is in 'B' or 'B' goes around 'A')
Classic examples are 'without' and 'within'. These indicators tend to be fairly obvious.
'Reversal' - indicates that a sequence of letters should be written backwards to form all or part of the answer.
Again, these are usually quite clear (eg 'back', 'returns') but something like 'over' could be used too

You might find the odd other type cropping up, but these will cover most clues. Remember that in the case of hidden words, containers and reversals, they might refer not to the word which appears in the clue, but another word meaning the same thing, or of the same type. If there is no indicator for all or part of the cryptic part then the word (or an equivalent word) will simply form part of the answer as it is.

Note that the indicators can mostly be combined together in the cryptic part of the clue, so for example a single letter indicator could give you a letter which will be part of the anagram, or there could be a reversal indicator and a container indicator so that one word goes backwards inside another one.

OK, I now realise that I have probably further confused you rather than helping you to understand how to do cryptic crosswords, but don't give up! Have a go at a cryptic crossword, bearing what I've said in mind - don't read the clue as one whole sentence - try to split it up, try to spot the indicators (remembering that they could be before or after the word they refer to). If you think you've spotted an anagram indicator, find the letters you think the answer is an anagram of, try writing them in a circle - many people find this helps spot the anagram. And remember, when the solution is available, go through the answers you didn't get and notice particularly the indicators that you hadn't spotted.