Modern CPUs support something called dual mode operation. CPUs with this capability use two modes: protected mode and supervisor mode, which allow certain CPU functions to be controlled and affected only by the operating system kernel. Here, protected mode does not refer specifically to the 80286 (Intel's x86 16-bit microprocessor) CPU feature, although its protected mode is very similar to it. CPUs might have other modes similar to 80286 protected mode as well, such as the virtual 8086 mode of the 80386 (Intel's x86 32-bit microprocessor or i386).
However, the term is used here more generally in operating system theory to refer to all modes which limit the capabilities of programs running in that mode, providing things like virtual memory addressing and limiting access to hardware in a manner determined by a program running in supervisor mode. Similar modes have existed in supercomputers, minicomputers, and mainframes as they are essential to fully supporting UNIX-like multi-user operating systems.
When a computer first starts up, it is automatically running in supervisor mode. The first few programs to run on the computer, being the BIOS, bootloader and the operating system have unlimited access to hardware. However when the operating system passes control to another program, it can place the CPU into protected mode.
In protected mode, programs may have access to a more limited set of the CPU's instructions. A user program may leave protected mode only by triggering an interrupt, causing control to be passed back to the kernel. In this way the operating system can maintain exclusive control over things like access to hardware and memory.
The term "protected mode resource" generally refers to one or more CPU registers, which contain information that the running program isn't allowed to alter. Attempts to alter these resources generally causes a switch to supervisor mode.