Parents, the safest way to keep your kids from accidentally spending your money in any of the iTunes stores, including the App Stores, is to give them their own iTunes Account that doesn't have an associated credit card.
Whether you want to let your kids play around safely, or just don't have a credit card, you can use an Apple Account with no credit card to try one of the hundred free apps in the new Mac App Store or download free apps on an iPhone or iPod touch.
You can't create an account in the main iTunes Store without a credit card, but you can from the App Stores.
From iTunes, if you're already signed into an account, sign out. Choose your country (use the country flag at lower right), then navigate into the App Store section using the top nav bar. Now, you must try to "buy" a free app, such as Apple's iTunes Remote. You'll be prompted to Create a New Account. Go through the process, making sure to choose "None" for your payment option. Check your email and click the link to verify your account. The process is essentially the same from an iPhone, iPad or iPod touch, but you'll specify your country after you click Buy.
Now you have an iTunes Store Account without a credit card on file, and you don't have to worry about your little snowflakes raiding your wallet for the latest holiday edition of Angry Birds.
We found this tip in the new Mac App Store Support site. For a step by step guide, along with instructions for setting up the free account from an iPhone, iPad, or iPod touch, see Apple's Knowledge Base article.
UPDATE: In the comments, some readers are saying this is not working for them. I've confirmed on the iPhone, in iTunes, and even in the Mac App Store itself: as of 10 PM UTC on 7 January 2011, this tip still works as described. Sign out any existing ID. Find a free app, click to buy, and follow the steps to create a new account. For your Credit Card type, choose None, exactly as shown in the screenshot above for iTunes, the button labled "None" in the Mac App Store, or the last item in the scrolling menu on your iPhone or iPod touch.