My wife got an e-mail that says a purchase was made at the app store. How can I cancel this?

If all you got was an email, see this:

Apple will not address you as Customer, they use your name provided when you created your Apple ID.

If all you received was an email, it is very possible this is a phishing attempt to get your account details. Do not interact or click on any links in the email and see this. Avoid phishing emails, fake 'virus' alerts, phony support calls, and other scams - Apple Support

This will help identify legitimate emails from Apple, Identify legitimate emails from the App Store or iTunes Store - Apple Support

Next, check your purchase history to see if there were actual purchases, See your purchase history in the App Store or iTunes Store - Apple Support

If there were not, just ignore the mail, send to Apple as an attachment to reportphishing@apple.com

If there were, and you see a deduction in your bank/credit card account, then see this, If you see 'itunes.com/bill' or an unfamiliar charge on your statement - Apple Support

If you are seeing charges for a subscription, see this for viewing and managing the subscription. View, change, or cancel your subscriptions - Apple Support

Did the e-mail receipt addresses you with your Full Name?

If it's Dear Customer, Dear Apple User, Dear Client, Dear your-email-address or something similar then it's 100% scam.

If the receipt is real, please note: all purchases are final.

You can plead your case using the Report a Problem link in the receipt.

See this reply that was made in a new thread: Response to your answers.

Thanks Fox, I told him to continue on this thread and not starting a new one.