In unix, every file and folder has permissions that basically determine who can Read, Write, or eXecute a file, be it root, administrator, or general user.
Often system files belong to the root or administrator. If you change these files as you (they will ask for a password), they will get written with permissions belonging to your user. Occasionally, the system will read the file and ignore them if they don't belong to the system, I suppose for security.
Repair permissions is a typical repair process that searches the database and the system for permissions that are incorrect and repair them. It's dubious how effective that is, but for certain things it's crucial. For example, when I was editing the kernel extensions to allow four finger gestures in 10.5, I would save the ext file with my admin name, and if I didn't repair the permissions, on boot, the file would not be loaded (meaning the trackpad was disabled :lol).