"Verification Required" when installing free apps?

It is indeed possible to create an Apple ID without a credit card, see "Creating an iTunes Store, App Store, iBookstore, and Mac App Store account without a credit card" at http://support.apple.com/kb/ht2534.

If you entered credit card information in the past you can dissociate it from your Apple ID as follows.

On an iOS device

1. Open Settings.

2. Tap iTunes & App Stores:

   

3. Tap your account and select "View Apple ID".

4. Select Payment information:

   

5. Select None:

   

6. Confirm with Done.

On a Mac

1. Open App Store.app:

   

2. Select menu Store>View My Account.

3. In the Apple ID Summary section locate the Payment Information line and click Edit >:

   

4. Select None as payment method:

   

   

5. Click Done to confirm.

If you don't see a "none" option, you may need to disable family sharing and then try the above steps again.