The apostrophe is used in writing contractions — that is, shortened forms of words from which one or more letters have been omitted. In standard English, this generally happens only with a small number of conventional items, mostly involving verbs. Here are some of the commonest examples, with their uncontracted equivalents:
It is not wrong to use such contractions in formal writing, but you should use them sparingly, since they tend to make your writing appear less than fully formal. Since I'm trying to make this document seem chatty rather than intimidating, I've been using a few contractions here and there, though not as many as I might have used. But I advise you not to use the more colloquial contractions like she'd've in your formal writing: these things, while perfectly normal in speech, are a little too informal for careful writing.
Such contractions represent the most useful job the apostrophe does for us, since, without it, we would have no way of expressing in writing the difference between she'll and shell, he'll and hell, can't and cant, I'll and ill, we're and were, she'd and shed, we'll and well, and perhaps a few others.
A few words which were contractions long ago are still conventionally written with apostrophes, even though the longer forms have more or less dropped out of use. There are so few of these that you can easily learn them all. Here are the commonest ones, with their original longer forms:
Contractions must be carefully distinguished from clipped forms. A clipped form is a full word which happens to be derived by chopping a piece off a longer word, usually one with the same meaning. Clipped forms are very common in English; here are a few, with their related longer forms:
Important note: Contractions must also be carefully distinguished from abbreviations. Abbreviations are things like Mr for Mister, lb. for pound(s), bc for before Christ and e.g. for for example.
Finally, there are a few circumstances in which apostrophes are used to represent the omission of some material in cases which are not exactly contractions. First, certain surnames of non-English origin are written with apostrophes: O'Leary (Irish), d'Abbadie (French), D'Angelo (Italian), M'Tavish (Scots Gaelic). These are not really contractions because there is no alternative way of writing them.
Second, apostrophes are sometimes used in representing words in non standard forms of English: thus the Scots poet Robert Burns writes gi' for give and a' for all. You are hardly likely to need this device except when you are quoting from such work.
Third, a year is occasionally written in an abbreviated form with an apostrophe: Pío Baroja was a distinctive member of the generation of '98. This is only normal in certain set expressions; in my example, the phrase generation of '98 is an accepted label for a certain group of Spanish writers, and it would not be normal to write *generation of 1898. Except for such conventional phrases, however, you should always write out years in full when you are writing formally: do not write something like *the '39–'45 war, but write instead the 1939–1945 war.