As a test launch iPhoto with the Option key held down and create a new, test library. Import some photos and test to see if the same problem persists. Does it? (This tells us if the problem is limited to your current library or is more wide spread)
If the new library allows you to click and drag then the culprit is your current library. In that case launch the library with the Option+Command keys held down and: 1 - repair the library permissions; 2 - rebuild the library database (will take two separated launchings).
If the problem persists in the new, test library log into another user account on your Mac and rerun the test there. This will tell us if the problem is limited to your user account or is system wide.
If system wide boot into Safe Mode and test your current library from there. Reboot normally and try again.
NOTE: Safe Mode boot can take up to 10 minutes as it's doing some system cache cleaning, volume verifying and directory repairing.