What are post-installation triggers? These are executed when upgrading software.
A dpkg trigger is a facility that allows events caused by one package but of interest to another package to be recorded and aggregated, and processed later by the interested package. This feature simplifies various registration and system-update tasks and reduces duplication of processing.
A more detailed description of dpkg triggers is here: http://lists.debian.org/debian-dpkg/2007/04/msg00076.html
Debian packages contain more than just files. Among other things, they often contain scripts that doesn't get installed, but are run before or after installing/uninstalling the files from a package. For example, the linux-kernel packages will run a script that adds the newly installed kernel to the bootloader menu after it has installed the actual files.
So when it says post-installation triggers, it is running such scripts.