These are all different acronyms for broadband remote access servers. A BRAS routes traffic going in and out of broadband remote access devices such as digital subscriber line access multiplexers, DSLAM for short.
The different acronyms are a result of the early Internet era when naming conventions had yet to be standardized. In fact, BRAS, B-RAS, and BBRAS are sometimes also called broadband network gateways or border network gateways.
If you’re interested in learning more about broadband remote access servers, the DSL Forum TR-025, TR-059, and TR-101 outline the architecture and structure of how BBRAS are made and used.