In its simplest definition,
A keylogger is something that intercepts signals between the keyboard and the OS proper. This is not generally difficult because the I/O required is published and easily accessible. You can have both hardware devices (plug the keyboard into the keyloggeer which then plugs into the system) or software, and you hope that the OS has sufficient security policy in place to prevent rogue, unauthorized installation. This is why you should never do anything related to your Personal Information on a shared system, always use your own, personal hardware.
My most recent amusing encounter with a keylogger is the extremely popular alternative Google Keyboard installable on any Android device through the Play Store which provides a full keyboard instead of a custom “one hand” keyboard/keypad. In the Terms of Agreement Google is very up front and transparent about how the use of the keyboard sends everything to Google which “may” be used as they wish.
So, not all keyloggers are criminal or under-handed, it can also be with complete disclosure.
TSU