Usage and example of the word “litotes”

Ward Farnsworth gives this definition:

Litotes (lye-tuh-teez) occurs when a speaker avoids making an affirmative claim directly and instead denies its opposite. Often this amounts to a double negative.

He gives a whole host of examples in his book Classical English Rhetoric. Here is one using the double negative:

Thus I consent, sir, to this Constitution, because I expect no better, and because I am not sure that it is not the best — Ben Franklin, at Federal Ratifying Convention (1787)

However, litotes does not have to involve a double negative. It may often simply refer to faint praise, "the most the speaker can offer":

She was not quite what you would call refined. She was not what you would call unrefined. She was the kind of person that keeps a parrot. — Mark Twain, Following the Equator (1897)

It is often used simply as understatement, "a useful tool for indicating small amounts, for making a show of modesty, or for creating a tone of allowance."

"You'll be quite safe now," the curate was saying in the adjoining room, not without a touch of complacent self-approval such as becomes the victor in a battle of wits. — P. G. Wodehouse, A Damsel in Distress (1919)

Let's take something more recent now, from a car campaign by General Motors trying to revive a failing brand by using litotes in an ad slogan to suggest a heroic understatement:

This is not your father's Oldsmobile.

Meaning? What you thought was a stodgy brand is now thoroughly revamped and revived, enough so that young people will find it exciting. The "not your father's X" (or grandfather's X, etc.) slogan has become a familiar trope, and is used in many similar constructions nowadays.

Addendum It is worth noting that George Orwell famously had a problem with the "not un-" construction. Your mileage may vary.