When you open a contact's vCard in Address Book and click the "Edit" button, you should be able to edit the user's photo by double-clicking on it and either use the computer's webcam to snap a shot or choose a picture file and then perform some simple zooming and cropping on the photo. Some people are having problems where this behavior will not work, either on some or all of their cards.
If this problem happens for all of your contacts, first try removing the preference files for Address Book. These files are called "com.apple.addressbook.plist" and "com.apple.addressbook.adb.plist" and are located in the /username/Library/Preferences/ folder. Move these files to the desktop and then relaunch Address Book. Keep in mind you may have to add LDAP directories and other custom settings again.
If the problem happens only for a few contacts, then you can try removing the preference files but it is more likely the vCard itself has a problem and needs to be replaced. You can do this for a specific card by right-clicking it, choosing "Export vCard" and saving it to a temporary location. Then delete the contact and import the saved vCard. You can also just create a new contact card, copy your information to it, and remove the old one.
If there are many contacts that are corrupted, you can export the whole contacts list by using the "Address Book Archive..." option in the File → Export menus to save the entire contacts database. From here, remove all your contacts and then import the archive again using the same option in the File → Import menu.