Debian-based distributions are distributions that are a variant of Debian or one of its descendants. You can find an ancestor-list of Linux distributions here: List_of_Linux_distributions#Family-Tree_distribution_listing.
Some prominent Debian-based distributions are
Debian-based distributions differ from other distributions by their software package system. They use dpkg while Red Hat and Suse based distributions use RPM, Slackware uses tgz and Gentoo uses the source.