Auger always related to tools and engineering. Some augers are drill bits, others are used to bore holes in wood or ice, and others are used to unclog toilets. The word can also be a verb meaning to use an auger. Augur always relates to premonition. As a verb, it means (1) to predict or foretell or (2) to be an omen. As a noun, augur denotes something that is ominous or that serves as a bellwether.
Unsurprisingly, auger is often used in place of augur—for example:
But the fact the club was not even given a First Division licence would not auger well for its chances of overturning the decision at appeal. [Irish Times]
But anyone who glanced at the stat sheet saw another number that augered well for ODU. [The Virginian-Pilot]
The mistaken use of augur in place of auger is less common than the reverse error, perhaps because writers who use auger often have know-how when it comes to tools and engineering.
Examples
Twisting his ice auger—essentially a four-foot corkscrew—he drilled a new hole through the ice. [Wall Street Journal]
But in another way these patterns augur a bright future for an expansive American metropolis. [Forbes]
New equipment for the primary sewer treatment centre, including an auger and grinder, were approved. [Quesnel Observer]
This does not augur well for either Mr. Huntsman or Mr. Romney, social moderates despite their Mormon faith. [Globe and Mail (article now offline)]
Which is why I was augering a hole through the icy surface of Coal Lake on a sunny Monday afternoon in February. [Edmonton Sun (article now offline)]
But, with due hesitation, I see changes that could augur a new era, one in which infantilized Arabic speakers mature into adults. [National Review]