The different modes for mobile device management (MDM) include BYOD, corporate-owned, and kiosk modes for Android, while iOS has managed, supervised, and dedicated modes. These modes provide varying levels of control and security for mobile devices depending on the needs of the organization. 

BYOD mode allows employees to use their own devices for work purposes while corporate-owned mode provides full control over devices owned by the organization. Kiosk mode is used for devices that are intended for a specific purpose or single-use, such as digital signage. Similarly, managed mode for iOS allows organizations to manage certain aspects of the device while preserving employee privacy, while supervised mode provides complete control over the device, and dedicated mode is designed for single-use devices.

The choice of MDM mode depends on the specific needs and requirements of the organization, as well as the operating system and device being used.