It basically consists of the small /etc/syslogout shell script which invokes other small shell scripts having a .bash suffix which are contained in the /etc/syslogout.d/ directory. The system administrator can drop in any script he wants without any naming convention other than that the scripts need to have a .bash suffix to enable automagic sourcing by the /etc/syslogout script.
For shell sessions, the contents of /etc/syslogout.d/ will be sourced by every user at logout if the following lines are present in his $HOME/.bash_logout:
if [ -f /etc/syslogout ]; then
. /etc/syslogout
fi
If used for X sessions it is advisable to include the former statement into the Xreset script of the X display manager instead to prevent that closing of an terminal emulator window yields unexpected results in your running X session if your X11 terminal emulator is using a login shell. Be sure then to run it under the user-id of the X session's user. See the example files in /usr/share/doc/syslogout/ for illustration.
Users not wanting /etc/syslogout to be sourced for their environment can easily disable it's automatic mechanism. It can be disabled by simply creating an empty file called $HOME/.nosyslogout in the user's home directory using e.g. the touch(1) command.
Any single configuration file in /etc/syslogout.d/ can simply be overridden by any user by creating a private $HOME/.syslogout.d/ directory which may contain a user's own version of any configuration file to be sourced instead of the system default. It's names have just to match exactly the system's default /etc/syslogout.d/ configuration files. Empty versions of these files contained in the $HOME/.syslogout.d/ directory automatically disable sourcing of the system wide version.
Naturally, users can add and include their own private scripts to be automagically executed by /etc/syslogout at logout time.
If you need a similar mechanism for executing code at login time check out the related package sysprofile(8) which is a very close companion to syslogout.