Business rules in Microsoft Dynamics CRM 2013 and CRM Online allow users to apply form logic without writing JavaScript code. Business rules provide a declarative interface to implement commonly used business rules through conditions and actions like setting field values, validation errors, and controlling field visibility and requirements. Some key capabilities of business rules include applying logic on form load and field changes, transporting between organizations, and localizing error messages. However, business rules have limitations compared to custom scripts, including an inability to use complex logical operators like OR in conditions.