OpenBOR is the open source continuations of Beats of Rage. This means that a group of fans are now creating new versions of Beats of Rage, based on the Beats of Rage source code which was released by Senile Team long ago. OpenBOR provides additional functionality not available in the original Beats of Rage game. It allows modders to add more moves and combos, amongst other things.
Note: the OpenBOR itself is the game engine, based on Beats of Rage engine (latter is not developed anymore). The games are installed as mods (for example, you can install Beats of Rage mod, which gives you original BOS game). The original game mod is named "Beats of Rage (The original game)". You'll have to register at Lavalit website to download any files.