The apostrophe or inverted comma is a much misused part of English punctuation, but there is a prescribed way of using it, which is quite Byzantine even when the rules are set out. As with so much of English, it is somewhat illogical in its function, the finer points being confused even by master grammarians. One way of looking at it is that the apostrophe is always used to mark contractions - spaces in words where letters have been left out for ease of pronunciation. However, because of the organic way in which English has evolved, there are plenty of loopholes, to be clarified here.
The language and attitudes towards it are evolving. The rapid evolution of computers has led to a general reduction in the amount of punctuation used. What was convention 30 years ago is now largely ignored. Formerly, plane was regarded as a contraction of aeroplane, and used to be written 'plane. No one does this any more, and because plane has become a word in its own right; most grammarians agree that this usage is no longer regarded as correct. Thus, while this article dwells on 'rules', it should be remembered that they are not set in stone for all time.
Where to Use Apostrophes
The apostrophe is used to denote missing letters in words that over the years have been dropped to ease pronunciation. The commonest examples are the auxiliary verbs and the word 'not'. Thus I am becomes I'm and are not becomes aren't. Of course there are exceptions, notably am not changing to ain't in what is a stupendous piece of illogicality1. Some nouns can be shortened in the same way: a ship's forecastle proved to be too hard to say, so the second syllable was dropped and it became fo'c'sle because lazy sailors preferred it that way. Similarly, five of the clock became five o'clock. Some words gain apostrophes only in lyrical works, to help illustrate the rhythm - an examples of this is ev'ry.
Possession is nine tenths of the law, even in the grammar of apostrophes. The majority of pitfalls in using apostrophes occur when trying to say something belongs to something else. But with the aid of this handy guide, grammatical foibles will be eradicated forever. How handy.
Singular possessive: The apostrophe is used to indicate possession with all nouns, both proper and common. The rules here are straightforward: add an apostrophe and an s. The ball belonging to Jack is Jack's Ball. The ball belonging to the man is the man's ball. The apostrophe comes between the word and the s. What could be easier?
However, words ending with an s present a problem. There are two valid options in this case. It is either Jesus' teaching or Jesus's teaching. Elegance would seem to preclude the latter.
Plural possessive: The same elements are used, but in a different order: add an s then an apostrophe. Thus the ball belonging to the boys is the boys' ball.
Naturally, the English language is not going to let such a rule getting away with being so easy. For plural nouns not ending in s, like men, for example, the rule is the same as for most singular nouns. The ball belonging to the men is the men's ball (Manchester United, for example, has lots of balls) .
Plurals (see below!)
In the solitary case of single lower case letters, it is preferable to use an apostrophe to avoid confusion, as in 'mind your p's and q's'.
An apostrophe also appears in foreign words written in English, used to transcribe a curious pronunciation, the glottal stop2: Qur'an, Hawai'i. The apostrophe in these cases is purely decorative, as the glottal stop is usually ignored by English speakers.
Where Not to Use Apostrophes
Possessive Adjectives and Pronouns
These are technical terms for two very common groups of words. The first group, is your, her, his, its and their. Notice the complete lack of apostrophes - the ball belonging to it is signified as its ball. And 'his' is not 'he's'. The corresponding possessive pronouns all end in s, with, once again, not an apostrophe in sight: yours, hers, his, its and theirs.
There is perhaps understandable confusion between the 'its' here with the contraction of 'it is' - it's. This is somewhat contradictory but helps to show that there is a difference in meaning between it is and that which belongs to it. This rule is frequently overlooked. Don't be a statistic.
Excepting the one case above, there is no need to use any apostrophe with plurals.
There seems to be some primordial urge in writers of shop signs to use it to denote plurals. This is manifestly wrong.
CD's: The plural of CD is, by convention, CDs. Alternatively, CD's may be used too, though this is not at all recommended. This applies to all other abbreviations like HGVs or A to Zs. Capisce?
1000's: If by some gruesome mischance you feel you need to butcher the beauty of our language by telling us you have lots of things on offer in, say, your Poundsaver store, please have the decency to write 1000s. Especially if you have 1000s of CDs
The 70's: The 70's are in fact the '70s. And don't you forget it!
Pencil's: That means belonging to the pencil, not more than one pencil.
A Note on Style
Although clearly with the knowledge of correct apostrophe usage burning in your veins you will want to put your new-found skills to good use, be aware that, as with everything, there is a time and a place. In a formal essay or letter, using a word like don't is very poor style indeed. Choose the elongated form.
There are many factors to be taken into consideration when making a quotation. Punctuation varies in this case between the US and Great Britain, and printers' conventions are different again.
In British English: Quotations are displayed with a single inverted comma: 'Thou, Nature, art my goddess; to thy law my services are bound'3. Speech marks are double inverted commas, like this ".
In American English: Here, conventions dictates that both quotation and speech are displayed using double inverted commas. Increasingly, British printers choose to use the American convention in order to appear more global.
In General: One final point. To quote speech, do this: '"Your punctuation is atrocious," he said'.