The easiest way to determine the "basis" of a distribution is the type of package management.
Like Jenni said.
For example RPM packages characterize Red Hat based systems.
-.deb (Debian) packages characterize Debian based systems-
Slackware has it's own tools but unlike RPM and Debian systems does not do automatic dependency resolution.I (think) Arch linux uses 'pacman'
Debian's commandline utility is 'apt' while Red Hat & Fedora's commanline utility is yum:-
Dependency is a program or a library that another program or application needs to run.
They (cmdline utility) both install, update, and remove packages.
The other differences are the desktop envirnments like gnome,unity,xfce and kde.
Wish you the best in your studies!