As our writing becomes increasingly colloquial (think texting), contractions are showing up more often in print. And these days, only the stuffiest style guides outlaw contractions altogether. Most, like the AP Stylebook, simply warn against "excessive use" in more formal kinds of writing.
If for some reason you're troubled by this recent proliferation of contracted forms, be glad you didn't live in the 17th century. At that time, in the words of editor William Gifford, printers habitually "deformed" texts with "barbarous contractions," lopping off letters without much regard for consistency or sense.
To show there's some degree of logic to current ways of contracting words, we've put together a few notes. . . .
For the complete article (revised and expanded), see Notes on Contractions in English.