Protected mode, on the 80286 and later members of the Intel x86 line, refers to a mode of the CPU in which it enables the extended architecture of the later chips, including hardware support for virtual memory.
These chips power on in so-called real mode, in which they appear to be 8086's; this is for purposes of backward compatibility at the object code level. A small extension to that architecture allows them to switch into protected mode, where all the extended features of the later architecture are available.
Note: When an x86 CPU is in protected mode, than does not mean that one program running on the CPU necessarily is protected from another program, merely that it can be, if the operating system sets up the memory management, etc properly.
Protected mode in the 286 is somewhat different from that of the 80386 and later machines, although the architecture of 286 protected mode is almost entirely upwardly compatible to that of the 386.