The files directory: Troubleshooting Apache-based Web Servers
If the directory files is not "owned" by the Web server there are two ways to proceed. These instructions assume you have shell access to the Web server. If you do not have shell access, please contact your hosting provider for additional support.
Ideally you will be able to change the "owner" of the directory files to match the name of the Web server. On a Unix or Linux-based server you may verify "who" the Web server is running as by issuing the following commands:
$ ps aux |grep apache # for Apache 2.x
$ ps aux |grep httpd # for Apache 1.x
Depending on your Web server one of these commands will return a series of lines like this:
www-data 13612 0.1 0.9 50640 20340 ? S 12:29 0:08 /usr/sbin/apache2 -k start
The first column in the returned text is the "user" that your Web server is operating as. In this case the user is www-data. To make your files directory writeable by the Web server you need to reassign its ownership using the following command:
$ chown -R www-data sites/default/files
$ chmod -R 0700 sites/default/files
If you do not have sufficient permissions on your server to change the ownership of the folder, the next best option is to provide group-write access to the folder with the following command:
$ chmod -R 0770 sites/default/files
You should also be able to adjust the directory's permissions using an FTP program as outlined above.
Once you've completed these steps confirm the appropriate permissions have been assigned by navigating to Administer -> Reports -> Status report. Scroll to the section labelled, "File system" and confirm your settings are now correct.
Source: The files directory