UNIX provides a number of standard logging facilities. These record information about various kinds of events on your system, which can be used to trace user activity, collect general statistics, and debug problems.
While such facilities are highly useful to the system administrator, it is also important to remember that there are ethical and security concerns about log information. Collecting information on user activity may be considered an invasion of their privacy under certain circumstances, and in general one should disclose to users that their activities are being logged, perhaps even detailing which things are logged. Log information may also be used by users to spy on other users if the log data is generally readable. It is good practice to make system logs accessible only to authorized administrators, and to limit disclosure of log data. Some logs may also contain (deliberately or inadvertently) security-critical information like passwords, and hence should also be carefully protected.
Unfortunately UNIX log information is not in a standardized format and there are many application-specific log files. While the syslog facility does provide a central standard logging mechanism, it is highly configurable as to where it places log data and whether it logs data about certain events at all. Some other log facilities tend to have fairly standard locations and utilities for processing the data, while application-specific logs (such as web server logs) are also configurable in both their location and output format.