Brine is fundamentally a salty solution. At home, it is used in pickling vegetables or preparing meat for cooking. This type of brine can be easily flushed down the sink and won't affect the plumbing or the septic system. However, large quantities of brine that are designed for industrial use or de-icing sidewalks can affect the water quality. Some used brine will have metal byproducts in it that is harmful or the high salt levels could affect the drinking water. It is necessary to dispose of this brine carefully so you won't have an adverse impact on the local environment.
Call your local city hall or health or environmental department to find out the regulations for disposing of brine. The regulations will vary depending on the quantity and what the brine was used for.
Take the brine to the local hazardous material handling facility if required and inform the staff that you wish to dispose of it.
Follow the staff's instructions. An injection well will probably be used to dispose of the brine. This system pushed the brine well underground so it doesn't affect the drinking water.