Many companies need to make lots and lots of decisions in a day, based on various factors. Each decision is governed by set of rules. Rules change = Decision change. These decisions are very critical for a company and hence there is a need to manage these rules which leads to the birth of BRMS.
Like for example: An insurance company has to make a lot of decisions on whether or not to provide a insurance policy to an insurer. Underwriting is a process by which an underwriter assess various factors and risks involved in providing the insurance to the insurer. The job of Underwriter involves a heavy use of BRMS behind the scenes. BRMS is also heavily used in banking industry for many decision making processes.
In short BRMS is a rules management system used to maintain, deploy, execute rules through out any organization for decision making.
One nice product that comes to my mind is by Fair Isaac Corporation(FICO) is FICO Blaze Advisor which is used by many banking and insurance companies around the globe. For more information here is their product page http://www.fico.com/en/Products/DMTools/Pages/FICO-Blaze-Advisor-System.aspx.