I guess I should have updated the CKEditor (currently 7.x-1.8), Date (7.x-2.3) & Webform (7.x-3.17) modules before updating Drupal 7 core. Now, I am unable to update the three modules. When I go to Home » Administration » Reports » Available updates, and then check any one or all three modules, I click Download the Update. Then, I see "Ready to update" and then after clicking "Continue", the site goes directly back to the home page. Any suggestions of how I might be able to update my three modules?
This is probably a problem with folder/file permissions. You can try changing the permissions of the sites/all/modules folder, and of the problematic modules' folders. You can read more about changing permissions here (follow the links there to see what's recommended; basically 755 should do). Possibly these modules cannot be removed by Drupal because of the ownership settings of the files/folders in question.
You can also do a manual update, as explained in the answers to this question.