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
Open Settings.
Tap iTunes & App Stores:

Tap your account and select "View Apple ID".
Select Payment information
:

Select None
:

Confirm with Done
.
On a Mac
Open App Store.app:

Select menu Store>View My Account.
In the Apple ID Summary section locate the Payment Information line and click Edit >
:

Select None
as payment method:


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.