How to update drupal 7.7 to 7.8 manually? I found that there is a warning on my site saying that manual updates of Drupal core is required.

How can I do it safely?
See this article from the Drupal.org documentation under section "MINOR VERSION UPDATES".


  To update from one minor 7.x version of Drupal to any later 7.x
  version, after following the instructions in the Introduction section
  at the top of this file:
  
  
  Log in as a user with the permission "Administer software updates".
  Go to Administration > Configuration > Development > Maintenance mode. Enable the "Put site into maintenance mode"
  checkbox and save     the configuration.
  Remove all old core files and directories, except for the 'sites' directory, the original install profile in the 'profiles'
  directory     and any custom files you added elsewhere.
  Download the latest Drupal 7.x release from http://drupal.org/project/drupal to a directory outside of your web 
  root. Extract the archive and copy the files into your Drupal
  directory.
  Re-apply any modifications to files such as .htaccess or robots.txt.
  Run update.php by visiting http://www.example.com/update.php (replace www.example.com with your domain name). This will update
  the    core database tables. 
  Go to Administration > Reports > Status report. Verify that everything is working as expected.
  Ensure that $update_free_access is FALSE in settings.php.
  Go to Administration > Configuration > Development > Maintenance mode. Disable the "Put site into maintenance mode" checkbox and save
  the configuration.