After a corp. action, an option's number of shares or its multiplier may be changed. What is the significance?

The option is adjusted to ensure fairness to both parties to the trade. If a stock splits 2 for 1, then it's only fair that the original option contract settles into 200 of the new shares instead of 100 of the old. Very often, the multiplier (usually 100) is also adjusted. When multiplied by the trade price, the multiplier determines how much actual cash changes hands during a trade. E.g., if an option is trading at $1.10, and the multiplier is 150, then I would have to pay $165.00 to buy one option contract. I would appreciate a worked example demonstrating how and why the multiplier is important when adjusting an option in response to a corporate action such as a stock split, a stock dividend, a spin-off, etc.