Syslog is an operating system administration logging tool, where messages are routed by facility and severity. It is more suitable for medium to large sites, and would not be suitable for shared hosting environments. In the file /etc/syslog.conf you define where messages go for any combination of facility and severity.
Here is a three-step guide to switching from Drupal's default database logging to Syslog, modified and expanded from http://mattdanger.net/2011/02/setup-syslog-on-drupal-6-x-and-7-x/. It is for Linux systems only.
1) Install & configure the Syslog module
Go to the Modules page, /admin/build/modules, and install the Syslog module. Then go to the Syslog settings page, /admin/settings/logging/syslog, and select which Syslog level to attach to the log messages. Choose one that is not in use by Syslog
2) Configure Syslog to Log to a Separate File
Edit /etc/syslog.conf and add:
Then, restart Syslog:
service syslog restart
Note that this is optional. If you do not do this, your messages will most likely be in /var/log/messages.
3) Disable the Database Logging (formerly, Watchdog) Module
Check that you are seeing messages in the syslog by using
You can also browse the file by using less.
If it is working, you may now go to the modules page and disable Dblog.
For more information on syslog facilities, severity levels, and how to setup a syslog.conf file, see UNIX/Linux syslog.conf and PHP's syslog.