Can you explain how overallotment options are typically used?

Generally, the underwriter has the ability to purchase additional registered shares from the Company.  For example, assume the Company registers 1 mil shares and an additional 150,000 overallotment.  The underwriter could just oversell the offering at the beginning and take down the entire 1,150,000 on day one.  The underwriter could take down the initial 1 mil on day one and then if the price is going up and the offering has interest, it could keep selling even though it doesn't techically own the 150,000 yet (technically going short) and then cover the short position by buying direct from the company at the IPO price.  The overallotment helps the underwriter continue to stablize the price of the stock, post closing.