It was the conference formed by the Texas schools and Arkansas, and lasted up until 1995 or so.
Southwest Conference - Wikipedia, the free encyclopedia
A lot of those stories you hear about texas, A&M, SMU (and its death penalty), TCU, etc, were from that period in time. It big enough that it would likely have also been a BCS conference had it lasted until 1998.
It was also rampant with cheating, corruption, etc, amongst the schools. That's what eventually led to it folding.
After that happened the Big 8 reached out and offered membership to 4 of the schools (Texas, A&M, Texas Tech, and Baylor - which went out of its way, supposedly going behind people's backs - to make sure TCU nor SMU got that last invitation) and became the Big 12 and Arkansas moved to the SEC.