What is a contraction?

A word is said to be contracted when it is shortened, in which case certain letters or sounds are omitted. An apostrophe is usually used to indicate the missing letter(s) in contracted words. Commonly contracted are auxiliaries, verbs or modals attached to other words. 
I'd = I had or I would. She'll = She will. He mustn't = He must not.  
Here are words and their contractions:
not = -n't
are = -'re
would, had = -'d
will = -'ll
is = -'s
I am  = I'm
Let us = let's

We are so familiar with contractions in speech that it's common to make mistakes when using them in writing. This is linked to the fact that contracted words are usually pronounced in a non-precise manner. For example, we pronounce should've, must've, they've in a way that it looks like we are saying should of, must of, they of, etc. These are mistakes and should never be used in writing since the contractions here mean should have, must have and they have respectively.

Other contractions:
There are other contractions that are less common but used by writers sometimes when representing a style of speech.  For example, you might have come across ne'er in American poetry, a contraction that's the shortened form of "never." Years in decades are also commonly contracted. So we have '30s '40s '50s etc. Some words are also contracted without apostrophes, e.g., lemme = let me gotta = have got a/ have got to.

When to use contractions:

You can't go wrong using contractions, but you'll want to limit their use to informal writing and not in formal writing as they confer a casual tone. However, there are rare cases where they become necessary in formal writing, for example, PM which is the contracted form of post meridiem. The contracted form can be used since the full word is rarely used in writing. 

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