Half precision floating point is a 16-bit binary floating-point interchange format. It was not part of the original ANSI/IEEE 754 Standard for Binary Floating-Point Arithmetic published in 1985 but is included in the current version of the standard, IEEE 754-2008 (previously known as IEEE 754r) which was published last August. See this Wikipedia article for background information.
Floating point formats defined by this standard are classified as either interchange or non-interchange. In the standard storage formats are narrow interchange formats, i.e. the set of floating point values that can be stored by the specified binary encoding is a proper subset of wider floating point formats such as the 32-bit float and 64-bit double. In particular, the standard defines the encodings for one binary storage floating-point format of 16 bits length and radix 2, and one decimal storage floating-point format of 32 bits length. Note that these two formats are for storage only and are not defined for in-memory arithmetic operations. The remainder of this post is about the 16-bit binary storage format which we will refer to as half precision.
Half precision (also known as a 1.5.10 or s10e5 minifloat) was added to the standard because it is the de facto storage format for certain floating-point values frequently used in modern graphics processing units (GPUs) where minimizing memory usage and bus traffic is a major challange and priority.& It is used in several computer graphics environments including OpenGL, OpenEXR, and by hardware in MP3 decoders and nVidia graphic cards.  This format became popular because it can store a larger range of values than an int16 without requiring the bandwidth and storage space of a float type. Typically this increased range of numbers is used to preserve more highlighting and shadow detail. The minimum and maximum representable values are 2.98×10-8 and 65504 respectively.
I only know of one C or C++ compiler which supports half precision i.e. Sourcery G++ Lite. It uses an __fp16 type to represent half precision with a number of limitations. However, whether __fp16 becomes part of ISO C remains to be seen.& The lastest version of the C++ ABI also provides some support for name mangling of half precisions. The GNU debugger appears to have some limited support. Ruby supports half precision (IEEE_binary16) using the float-formats package (but only for little endian platforms according to the float-formats README). The Python structs module which is the logical home for half-precision support does not currently support this format. A cursory search of CPAN did not