A mining rig is a computer system used for mining bitcoins. The rig might be a dedicated miner where it was procured, built and operated specifically for mining or it could otherwise be a computer that fills other needs, such as performing as a gaming system, and is used to mine only on a part-time basis.
Warning: GPU mining is not very profitable (if at all) anymore, and even if you have free electricity, GPU rigs will likely never pay for themselves at this point!